History
Malmö is located at 13°00' east and 55°35' north. Its location in southernmost Sweden makes it closer to the Italian city of Milan than to the northernmost Swedish town Kiruna.
Malmö is part of the transnational Oresund Region and since 2000 the Oresund Bridge crosses the Oresund strait to Copenhagen, Denmark. The bridge was inaugurated July 1, 2000, and measures 8 kilometres (the whole link totalling 16 km), with pylons reaching 204.5 metres vertically. Apart from the Helsingborg-Helsingør ferry links further north, most ferry connections have been discontinued.
Geography
The shores of Scania, where Malmö is situated, have a temperate climate and is according to Köppen climate classification part of the Maritime Temperate climates. This means that the average temperature is above 10 °C in the warmest months, and the coldest month average is above −3 °C.
Climate
Commuter trains pass the bridge every 20 minutes connecting Malmö to Copenhagen, and the Copenhagen Airport. Also some of the X2000 and Intercity trains to Stockholm, Gothenburg, and Kalmar pass the bridge. All these trains stop at Copenhagen Airport.
In March of 2005, digging began on a new railway connection called Citytunneln (The City Tunnel). The tunnel will run from under Malmö Central Station to Triangeln continuing to Hyllievång (Hyllie Meadow), where it will emerge to connect with the Oresund Bridge, effectively changing Malmö Central from being a terminus to being a transit station.
Beside the Copenhagen Airport, Malmö has an airport of its own, Malmö Airport, today chiefly used for low-cost carriers, charter flight routes, and domestic Swedish destinations.
The motorway system has been incorporated with the Oresund Bridge; the European route E20 goes over the bridge and then, together with the European route E6 follows the Swedish west coast from Malmö–Helsingborg to Gothenburg. E6 goes further north along the west coast and through Norway to the Norwegian town Kirkenes at Barents Sea. The European route to Jönköping–Stockholm (E4) starts at Helsingborg. Main roads in direction of Växjö–Kalmar, Kristianstad–Karlskrona, Ystad, and Trelleborg start as freeways.
Transportation
Monday, March 24, 2008
Sunday, March 23, 2008
Value added tax (VAT), or goods and services tax (GST), is tax on exchanges. It is levied on the added value that results from each exchange. It differs from a sales tax because a sales tax is levied on the total value of the exchange. For this reason, a VAT is neutral with respect to the number of passages that there are between the producer and the final consumer. A VAT is an indirect tax, in that the tax is collected from someone other than the person who actually bears the cost of the tax (namely the seller rather than the consumer). To avoid double taxation on final consumption, exports (which by definition, are consumed abroad) are usually not subject to VAT and VAT charged under such circumstances is usually refundable.
The VAT was invented by a French economist in 1954. Maurice Lauré, joint director of the French tax authority, the Direction générale des impôts, as taxe sur la valeur ajoutée (TVA in French) was first to introduce VAT with effect from 10 April 1954 for large businesses, and extended over time to all business sectors. In France, it is the most important source of state finance, accounting for approximately 45% of state revenues.
Personal end-consumers of products, consumers and services cannot recover VAT on purchases, but businesses are able to recover VAT on the materials and services that they buy to make further supplies or services directly or indirectly sold to end-users. In this way, the total tax levied at each stage in the economic chain of supply is a constant fraction of the value added by a business to its products, and most of the cost of collecting the tax is borne by business, rather than by the state. VAT was invented because very high sales taxes and tariffs encourage cheating and smuggling. It has been criticized on the grounds that it is a regressive tax.
Comparison with a sales tax
The standard way to implement a VAT is to say a business owes some percentage on the price of the product minus all taxes previously paid on the good. If VAT rates were 10%, an orange juice maker would pay 10% of the $5 per gallon price ($0.50) minus taxes previously paid by the orange farmer (maybe $0.20). In this example, the orange juice maker would have a $0.30 tax liability. Each business has a strong incentive for its suppliers to pay their taxes, allowing VAT rates to be higher with less tax evasion than a retail sales tax.
Collection Mechanism
Consider the manufacture and sale of any item, which in this case we will call a widget.
Example
A widget manufacturer spends $1 on raw materials and uses them to make a widget.
The widget is sold wholesale to a widget retailer for $1.20, making a profit of $0.20.
The widget retailer then sells the widget to a widget consumer for $1.50, making a profit of $0.30 Without any sales tax
With a 10% sales tax:
So the consumer has paid 10% ($0.15) extra, compared to the no taxation scheme, and the government has collected this amount in taxation. The retailers have not lost anything directly to the tax, but they do have the extra paperwork to do so that they correctly pass on to the government the sales tax they collect. Suppliers and manufacturers have the administrative burden of supplying correct certifications, and checking that their customers (retailers) aren't consumers.
The manufacturer pays $1.00 for the raw materials, certifying it is not a final consumer.
The manufacturer charges the retailer $1.20, checking that the retailer is not a consumer, leaving the same profit of $0.20.
The retailer charges the consumer $1.65 ($1.50 + 10%) and pays the government $0.15, leaving the same profit of $0.30. With a North American (Canadian Provincial and U.S. State) sales tax
With a 10% VAT:
So the consumer has paid 10% ($0.15) extra, compared to the no taxation scheme, and the government has collected this amount in taxation. The businesses have not lost anything directly to the tax, but they do have the extra paperwork to do so that they correctly pass on to the government the difference between what they collect in VAT (output VAT, an 11th of their income) and what they spend in VAT (input VAT, an 11th of their expenditure).
Note that in each case the VAT paid is equal to 10% of the profit, or 'value added'.
The advantage of the VAT system over the sales tax system is that businesses cannot hide consumption (such as wasted materials) by certifying it is not a consumer.
The manufacturer pays $1.10 ($1 + 10%) for the raw materials, and the seller of the raw materials pays the government $0.10.
The manufacturer charges the retailer $1.32 ($1.20 + $1.20x10%) and pays the government $0.02 ($0.12 minus $0.10), leaving the same profit of $0.20.
The retailer charges the consumer $1.65 ($1.50 + $1.50x10%) and pays the government $0.03 ($0.15 minus $0.12), leaving the same profit of $0.30. Limitations to example and VAT
The "value added tax" has been criticized as the burden of it relies on personal end-consumers of products and is therefore a regressive tax (the poor pay more, in comparison, than the rich). However, this calculation is derived when the tax paid is divided not by the tax base (the amount spent) but by income, which is argued to create an arbitrary relationship. The tax rate itself is proportional with higher income people paying more tax but at the same rate as they consume more. If a value added tax is to be related to income, then the unspent income can be treated as deferred (spending savings at a later point in time), at which time it is taxed creating a proportional tax using an income base. Such taxes can have a progressive effect on the effective tax rate of consumption by using exemptions, rebates, or credits.
Revenues from a value added tax are frequently lower than expected because they are difficult and costly to administer and collect. In many countries, however, where collection of personal income taxes and corporate profit taxes has been historically weak, VAT collection has been more successful than other types of taxes. VAT has become more important in many jurisdictions as tariff levels have fallen worldwide due to trade liberalisation, as VAT has essentially replaced lost tariff revenues. Whether the costs and distortions of value added taxes are lower than the economic inefficiencies and enforcement issues (e.g. smuggling) from high import tariffs is debated, but theory suggests value added taxes are far more efficient.
Due to the fact that exports are generally zero-rated (and VAT refunded or offset against other taxes), this is often where VAT fraud occurs. In sectors or countries where VAT fraud is prevalent, attempts by authorities to control fraud may have unintended consequences, and raise costs for honest companies. This problem is also true of other types of taxation, however.
Certain industries (small-scale services, for example) tend to have more VAT avoidance, particularly where cash transactions predominate, and VAT may be criticized for encouraging this. From the perspective of government, however, VAT may be preferable because it captures at least some of the value-added. For example, a carpenter may offer to provide services for cash (i.e. without a receipt, and without VAT) to a homeowner, who usually cannot claim input VAT back. The homeowner will hence bear lower costs and the carpenter may be able to avoid other taxes (profit or payroll taxes). The government, however, may still receive VAT for various other inputs (lumber, paint, gasoline, tools, etc) sold to the carpenter, who would be unable to reclaim the VAT on these inputs. While the total tax receipts may be lower compared to full compliance, it may not be lower than under other feasible taxation systems.
In Europe, the main source of problems is called 'carousel' fraud. Large quantities of valuable goods (often microchips or mobile phones) are transported from one member state to the other. During these transactions, some companies owe VAT, other acquire a right to reclaim VAT. The first companies, called 'missing traders' go bankrupt without paying. The second group of companies can 'pump' money straight out of the national treasuries.
This kind of fraud originated in the 1970s in the Benelux-countries. Today, the British treasury is the main victim. The British judicial system is considered by such criminals as the weakest in the EU. Collaboration with other member states is poor, and the trial-by-jury system makes convictions difficult.
Criticisms
VAT systems
A common VAT system is compulsory for member states of the European Union. The EU VAT system is imposed by a series of European Union directives, the most important of which is the Sixth VAT Directive (Directive 77/388/EC). Nevertheless, some member states have negotiated variable rates (Madeira in Portugal) or VAT exemption for regions or territories. The regions below fall out of the scope of EU VAT:
Under the EU system of VAT, where a person carrying on an economic activity supplies goods and services to another person, and the value of the supplies passes financial limits, the supplier is required to register with the local taxation authorities and charge its customers, and account to the local taxation authority for VAT (although the price may be inclusive of VAT, so VAT is included as part of the agreed price, or exclusive of VAT, so VAT is payable in addition to the agreed price).
VAT that is charged by a business and paid by its customers is known as output VAT (that is, VAT on its output supplies). VAT that is paid by a business to other businesses on the supplies that it receives is known as input VAT (that is, VAT on its input supplies). A business is generally able to recover input VAT to the extent that the input VAT is attributable to (that is, used to make) its taxable outputs. Input VAT is recovered by setting it against the output VAT for which the business is required to account to the government, or, if there is an excess, by claiming a repayment from the government.
Different rates of VAT apply in different EU member states. The minimum standard rate of VAT throughout the EU is 15%, although reduced rates of VAT, as low as 5%, are applied in various states on various sorts of supply (for example, domestic fuel and power in the UK). The maximum rate in the EU is 25%.
The Sixth VAT Directive requires certain goods and services to be exempt from VAT (for example, postal services, medical care, lending, insurance, betting), and certain other goods and services to be exempt from VAT but subject to the ability of an EU member state to opt to charge VAT on those supplies (such as land and certain financial services). Input VAT that is attributable to exempt supplies is not recoverable, although a business can increase its prices so the customer effectively bears the cost of the 'sticking' VAT (the effective rate will be lower than the headline rate and depend on the balance between previously taxed input and labour at the exempt stage).
Finally, some goods and services are "zero-rated". The zero-rate is a positive rate of tax calculated at 0%. Supplies subject to the zero-rate are still "taxable supplies", i.e. they have VAT charged on them. In the UK, examples include most food, books, drugs, and certain kinds of transport. The zero-rate is not featured in the EU Sixth Directive as it was intended that the minimum VAT rate throughout Europe would be 5%. However, zero-rating remains in some Member States, most notably the UK, as a legacy of pre-EU legislation. These Member States have been granted a derogation to continue existing zero-rating but cannot add new goods or services. The UK also exempts or lowers the rate on some products depending on situation; for example milk products are exempt from VAT, but if you go into a restaurant and drink a milk drink it is VAT-able. Some products such as feminine hygiene products and baby products (nappies etc) are charged at 5% VAT along with domestic fuel.
When goods are imported into the EU from other states, VAT is generally charged at the border, at the same time as customs duty. "Acquisition" VAT is payable when goods are acquired in one EU member state from another EU member state (this is done not at the border but through an accounting mechanism). EU businesses are often required to charge themselves VAT under the reverse charge mechanism where services are received from another member state or from outside of the EU.
Businesses can be required to register for VAT in EU member states, other than the one in which they are based, if they supply goods via mail order to those states, over a certain threshold. Businesses that are established in one member state but which receive supplies in another member state may be able to reclaim VAT charged in the second state under the provisions of the Eighth VAT Directive (Directive 79/1072/EC). To do so, businesses have a value added tax identification number. A similar directive, the Thirteenth VAT Directive (Directive 86/560/EC), also allows businesses established outside the EU to recover VAT in certain circumstances.
Following changes introduced on July 1, 2003, (under Directive 2002/38/EC), non-EU businesses providing digital electronic commerce and entertainment products and services to EU countries are also required to register with the tax authorities in the relevant EU member state, and to collect VAT on their sales at the appropriate rate, according to the location of the purchaser. Alternatively, under a special scheme, non-EU businesses may register and account for VAT on only one EU member state. This produces distortions as the rate of VAT is that of the member state of registration, not where the customer is located, and an alternative approach is therefore under negotiation, whereby VAT is charged at the rate of the member state where the purchaser is located.
The differences between different rates of VAT was often originally justified by certain products being "luxuries" and thus bearing high rates of VAT, whereas other items were deemed to be "essentials" and thus bearing lower rates of VAT. However, often high rates persisted long after the argument was no longer valid. For instance, France taxed cars as a luxury product (33%) up into the 1980s, when most of the French households owned one or more cars. Similarly, in the UK, clothing for children is "zero rated" whereas clothing for adults is subject to VAT at the standard rate of 17.5%.
Åland Islands (Finland)
Heligoland island, Büsingen territory (Germany)
Guadeloupe, Martinique, French Guiana, Réunion (France)
Mount Athos (Greece)
Ceuta, Melilla, The Canary Islands (Spain)
Livigno, Campione d'Italia, Lake Lugano (Italy)
Gibraltar, The Channel Islands (United Kingdom) European Union
Where most of the trade is business-to-consumer, displayed prices must include VAT and VAT must be charged.
Where most of the trade is business-to-business, displayed prices do not have to include VAT. For business transactions the following rules apply:
- VAT must be charged if the buyer is in the same country as the seller. The buyer may be able to reclaim the VAT from the tax authorities.
VAT does not need to be charged if the buyer is in a different country. The seller must record the VAT number of the buyer.
Certain EU companies are VAT exempt, these companies must not be charged VAT, regardless of whether they are in the same or different country to the seller. Rules on pricing within the EU
MOMS (Danish: Meromsætningsafgift, Norwegian: merverdiavgift (abriviated MVA), Swedish: mervärdesskatt, earlier mervärdesomsättningsskatt) is a Danish, Norwegian and Swedish sales tax. MOMS is the Danish, Norwegian and Swedish term for VAT. Like other countries' sales and VAT taxes, MOMS is a regressive indirect tax.
In Denmark, VAT is only applied at one level, and is not split into two levels as in other countries (e.g. Germany), where VAT is split into VAT for foodstuffs and VAT for nonfood. The current percentage in Denmark is 25%. That makes Denmark one of the countries with the highest value added tax, alongside Norway and Sweden.
In Norway, VAT is split into three levels: 25% is the general VAT, 14% (formerly 13%, up on January 1, 2007) for foods and restaurant take-out (food eaten in a restaurant has 25%), 8% for person transport, movie tickets, and hotel stays. Most printed matters are still free of VAT.
In Sweden, VAT is split into three levels: 25% for most goods and services including restaurants bills, 12% for foods and hotel stays (but breakfast at 25%) and 6% for printed matter, cultural services,and transport of private persons. Some services are not taxable for example education of children and adults if public utility, but education is taxable at 25% in case of courses for adults at a private school.
MOMS replaced OMS (Danish "Omsætningsafgift", Swedish "omsättningsskatt") in 1967, which was a tax applied exclusively for retailers.
Denmark, Norway, and Sweden (MOMS)
In India, VAT replaced sales tax on 1 April 2005. Of the 21 Indian states, eight did not introduce VAT. Haryana had already adopted it on 1 April 2004. The "empowered committee" of the basic framework for uniform VAT laws in the states. Due to the federal nature of the Indian constitution, the states do have the power to set their own VAT rate.
India
In the Indian state of Andhra Pradesh, the Andhra Pradesh Value Added Tax Act, 2005 came into force on 1 April 2005 and contains six schedules. Schedule I contains goods generally exempted from tax. Schedule II deals with zero rated transactions like exports and Schedule III contains goods taxable at 1%, namely jewellery made from bullion and precious stones. Goods taxable at 4% are listed under Schedule IV. The majority of foodgrains and goods of national importance, like iron and steel, are listed under this head. Schedule V deals with Standard Rate Goods, taxable at 12.5%. All goods that are not listed elsewhere in the Act fall under this head. The VI Schedule contains goods taxed at special rates, such as some liquor and petroleum products.
The Act prescribes threshold limits for VAT registration - dealers with a taxable turnover of over Rs.40.00 lacs, in a tax period of 12 months, are mandatorily registered as VAT dealers. Dealers with a taxable turnover, in a tax period of 12 months, between Rs.5.00 to 40.00 lacs are registered as Turnover Tax (TOT) dealers. While the former category of dealers are eligible for input tax credit, the latter category of dealers are not. A VAT dealer pays tax at the rate specifed in the Schedules. The sales of a TOT dealer are all taxable at 1%. A VAT dealer has to file a monthly return disclosing purchases and sales. A TOT dealer has to file a quarterly return disclosing only sale turnovers. While a VAT dealer can buy goods for business from anywhere in the country, a TOT dealer is barred from buying outside the State of A.P.
The Act appears to be the most liberal VAT law in India. It has simplified the registration procedures and provides for across the board input tax credit (with a few exceptions) for business transactions. A unique feature of registration in Andhra Pradesh is the facility of voluntary VAT registration and input tax credit for start-ups.
The Act also provides for transitional relief (TR) for goods on hand as of 1 April 2005. However, these goods ought to have been purchased from registered dealers between 1 April 2005 to 31 March 2006. This is a bold step compared to the 3 months TR provided by several developed countries.
The Act not only provides for tax refunds for exporters (refund of tax paid on inputs used in the manufacture of goods exported), it also provides for refund of tax in cases where the inputs are taxed at 12.5% and outputs are taxed at 4%.
The VAT Act in Andhra Pradesh is administered by the Commercial Taxes Department (department to collect VAT and other taxes) using a networked software package called VATIS. The personnel were trained prior to the Act coming into force. VATIS is used to process documents and forms received and to generate registration certificates and tax demand notices.
VAT, to be successful, relies on voluntary tax compliance. Since VAT believes in self assessments, dealers are required to maintain proper records, issue tax invoices, file correct tax returns etc. The opposite seems to be happening in India. Businesses are still run on traditional lines. Cash transactions are order of the day. The unorganised sector dominates the market. The hope of higher tax compliance and lesser evasion is still a far cry in Andhra Pradesh. This is reflected in the high percentage of return defaulters (14%), credit returns (35%) and nil returns (20%). That is, roughly 70% of VAT dealers are presently not paying any tax. Filing of credit returns is rampant among FMCG, Consumer Durables, Drugs and Medicines and Fertilizers. The margins are low in this sector (ranging between 2 to 5%). The value addition is not enough to yield revenue as of now. Credits offered by manufacturers compounds the problem. The question is, in a typical purchases and sales scenario, can there be more output tax than input tax? When purchases consistently exceed sales, can output tax exceed input tax? If a VAT dealer can balance his/her purchases and sales, can there be a net tax to the State? Is there a mathematical model or paradigm which can give value added tax and which can reduce the percentage of credit returns? There are no ready answers for these queries. The only remedy seems to be the restriction of input tax to the corresponding purchase value of goods put to sales. In fact a two tier system can be adopted to counter the credit returns - allow full input tax to manufacturers and restrict input tax to the purchase value of goods put to sale to traders. Restricting input tax to 4% in the case of inter state sales and in the case of products taxable at 12.5% seems to be another solution.
The Andhra Pradesh experience
Impuesto al Valor Agregado (IVA, "value-added tax" in Spanish) is a tax applied in Mexico and other countries of Latin America and Spain. In Chile it is called Impuesto a las Ventas y Servicios, in Spain Impuesto sobre el Valor Añadido and in Peru it is called Impuesto General a las Ventas or IGV.
Prior to the IVA, a similar tax called impuesto a las ventas ("sales tax") had been applied in Mexico. In September, 1966, the first attempt to apply the IVA took place when revenue experts declared that the IVA should be a modern equivalent of the sales tax as it occurred in France. At the convention of the Inter-American Center of Revenue Administrators in April and May, 1967, the Mexican representation declared that the application of a value-added tax would not be possible in Mexico at the time. In November, 1967, other experts declared that although this is one of the most equitable indirect taxes, its application in Mexico could not take place.
In response to these statements, direct sampling of members in the private sector took place as well as field trips to the European countries this tax was applied or it was soon to be applied. In 1969, the first attempt to substitute the mercantile-revenue tax for the value-added tax took place. On December 29, 1978 the Federal government published the official application of the tax beginning on January 1, 1980 in the Official Journal of the Federation (Diario Oficial de la Federación).
Mexico
Goods and Services Tax (GST) is a Value Added Tax introduced in New Zealand in 1986, which is currently 12.5%. It is notable for exempting few items from the tax...
New Zealand
Goods and Services Tax (GST) is a Value Added Tax introduced in Australia in 2000 which is collected by the Federal government but given to state governments.
Australia
In the United States, the state of Michigan uses a form of VAT known as the "Single Business Tax" (SBT) as its form of general business taxation. It is the only state in the U.S. to use a VAT. When it was adopted in 1975, it replaced seven business taxes, including a corporate income tax. On August 9, 2006, the Michigan legislature approved voter-initiated legislation to repeal the Single Business Tax. The repeal will be effective after December 31, 2007.. In many stores, the price tags and/or advertised prices do not include the taxes, these will be added at the cash register before the customer pays. In many states, no sales tax is charged for services. This is a key difference between most sales taxes levied throughout the United States and the value added taxes in other countries.
United States
Tax Rates
EU countries
Note 1: Some Canadian provinces collect 14% for harmonized sales tax, a combined federal/provincial VAT. In the rest, the federal GST is 6% and if the province charges sales tax it is separate and is not a VAT. No real "reduced rate" but rebates are generally available for new housing effectively reducing the tax to 4.5%
Note 2: These taxes do not apply in Hong Kong and Macau, which are financially independent as special administrative regions.
Note 3: The reduced rate was 14% until 1 March 2007, when it was lowered to 7%. The reduced rate applies to heating costs, printed matter, restaurant bills, hotel stays, and most food.
Note 4: VAT is not implemented in 2 of India's 28 states.
Note 5: The VAT in Israel is in the process of being gradually reduced. It was reduced from 18% to 17% on March 2004, to 16.5% on September 2005, and was set to its current rate on July 1, 2006. There are plans to further reduce it in the near future, but they depend on political changes in the Israeli parliament.
Note 6: In the 2005 Budget, the government announced that GST would be introduced in January 2007. Many details have not yet been confirmed but it has been stated that essential goods and small businesses would be exempted or zero rated. Rates have not yet been established as of June 2007.
Note 7: The President of the Philippines has the power to raise the tax to 12% after January 1, 2006. The tax was raised to 12% on February 1.
Note 9: The States of Jersey has for the few years, preparing for the introduction of a goods and sales tax to plug a large budget deficit in the island's government budget. It will be held at a flat rate of 3%, with possible exceptions to local food (food not subject to an "island tax" of 5%) and children's clothing.
Note 10: Except Eilat, where VAT is not raised.
Non-EU countries
VAT registered means registered for VAT purposes, i.e. entered into an official VAT payers register of a country. Both natural persons and legal entities can be VAT registered. Countries that use VAT have established different thresholds for remuneration derived by natural persons/legal entities during a calendar year (or a different period) by exceeding which the VAT registration is compulsory. Natural persons/legal entities that are VAT registered are obliged to calculate VAT on certain goods/services that they supply and pay VAT into particular state budget. VAT registered persons/entities are entitled to VAT deduction under legislatory regulations of particular country. The introduction of a VAT can reduce the cash economy because businesses that wish to buy and sell with other VAT registered businesses must themselves be VAT registered.
See also
Saturday, March 22, 2008
Friday, March 21, 2008
In mathematics, a series is often represented as the sum of a sequence of terms. That is, a series is represented as a list of numbers with addition operations between them, for example this arithmetic sequence:
1 + 2 + 3 + 4 + 5 + ... + 99 + 100.
In most cases of interest the terms of the sequence are produced according to a certain rule, such as by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator.
A series may be finite or infinite. Finite series may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way.
Examples of simple series include the arithmetic series which is a sum of an arithmetic progression, written as:
and finite geometric series, a sum of a geometric progression, which can be written as:
Mathematicians usually study a series as a pair of sequences: the sequence of terms of the series: a0, a1, a2, … and the sequence of partial sums S0, S1, S2, …, where Sn = a0 + a1 + … + an. The notation
represents then a priori this pair of sequences, which is always well defined, but which may or may not converge. In the case of convergence, i.e., if the sequence of partial sums SN has a limit, the notation is also used to denote the limit of this sequence. To make a distinction between these two completely different objects (sequence vs. numerical value), one may sometimes omit the limits (atop and below the sum's symbol) in the former case, although it is usually clear from the context which one is meant.
Also, different notions of convergence of such a sequence do exist (absolute convergence, summability, etc). In case the elements of the sequence (and thus of the series) are not simple numbers, but, for example, functions, still more types of convergence can be considered (pointwise convergence, uniform convergence, etc.; see below).
Mathematicians extend this idiom to other, equivalent notions of series. For instance, when we talk about a recurring decimal, we are talking, in fact, just about the series for which it stands (0.1 + 0.01 + 0.001 + …). But because these series always converge to real numbers (because of what is called the completeness property of the real numbers), to talk about the series in this way is the same as to talk about the numbers for which they stand. In particular, it should offend no sensibilities if we make no distinction between 0.111… and /9. Less clear is the argument that 9 × 0.111… = 0.999… = 1, but it is not untenable when we consider that we can formalize the proof knowing only that limit laws preserve the arithmetic operations. See 0.999... for more.
History of the theory of infinite series
The idea of an infinite series expansion of a function was first conceived in India by Madhava in the 14th century, who also developed the concepts of the power series, the Taylor series, the Maclaurin series, rational approximations of infinite series, and infinite continued fractions. He discovered a number of infinite series, including the Taylor series of the trigonometric functions of sine, cosine, tangent and arctangent, the Taylor series approximations of the sine and cosine functions, and the power series of the radius, diameter, circumference, angle θ, π and π/4. His students and followers in the Kerala School further expanded his works with various other series expansions and approximations, until the 16th century.
In the 17th century, James Gregory also worked on infinite series and published several Maclaurin series. In 1715, a general method for constructing the Taylor series for all functions for which they exist was provided by Brook Taylor. Leonhard Euler in the 18th century, developed the theory of hypergeometric series and q-series.
Development of infinite series
The study of the convergence criteria of a series began with Madhava in the 14th century, who developed tests of convergence of infinite series, which his followers further developed at the Kerala School.
In Europe however, the investigation of the validity of infinite series is considered to begin with Gauss in the 19th century. Euler had already considered the hypergeometric series
on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence.
Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long before by Gregory (1668). Leonhard Euler and Gauss had given various criteria, and Colin Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form.
Abel (1826) in his memoir on the binomial series
corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of m and x. He showed the necessity of considering the subject of continuity in questions of convergence.
Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Tchebichef (1852), and Arndt (1853).
General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's (from 1889) memoirs present the most complete general theory.
Convergence criteria
The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Seidel and Stokes (1847-48). Cauchy took up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thomae used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions.
Uniform convergence
A series is said to be semi-convergent (or conditionally convergent) if it is convergent but not absolutely convergent.
Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function
Genocchi (1852) has further contributed to the theory.
Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into prominence.
Semi-convergence
Fourier series were being investigated as the result of physical considerations at the same time that Gauss, Abel, and Cauchy were working out the theory of infinite series. Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète. Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and Kummer.
Fourier (1807) set for himself a different problem, to expand a given function of x in terms of the sines or cosines of multiples of x, a problem which he embodied in his Théorie analytique de la Chaleur (1822). Euler had already given the formulas for determining the coefficients in the series; Fourier was the first to assert and attempt to prove the general theorem. Poisson (1820-23) also attacked the problem from a different standpoint. Fourier did not, however, settle the question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner (see convergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and DuBois-Reymond. Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell.
Fourier series
In general, the geometric series
converges if and only if |z| < 1.
converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion described below in convergence tests. As a function of r, the sum of this series is Riemann's zeta function.
converges if the sequence bn converges to a limit L as n goes to infinity. The value of the series is then b1 − L.
A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example:
The harmonic series is the series
An alternating series is a series where terms alternate signs. Example:
The series
A telescoping series Some types of infinite series
Main article: absolute convergence.
A series
is said to converge absolutely if the series of absolute values
converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum.
The Riemann series theorem says that if a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Moreover, if the an are real and S is any real number, one can find a reordering so that the reordered series converges with limit S.
Absolute convergence
Main article: convergence tests Convergence tests
Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. For example, the series
converges to e for all x. See also radius of convergence.
Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required. However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.
Power series
Main article: Dirichlet series Dirichlet series
Asymptotic series, otherwise asymptotic expansions, are infinite series whose partial sums become good approximations in the limit of some point of the domain. In general they do not converge. But they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers.
Cesàro summation, (C,k) summation, Abel summation, and Borel summation provide increasingly weaker (and hence applicable to increasingly divergent series) means of defining the sums of series.
The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space.
Generalizations
Analogous definitions may be given for sums over arbitrary index set. Let a: I → X, where I is any set and X is an abelian topological group. Let F be the collection of all finite subsets of I. Note that F is a directed set ordered under inclusion with union as join. We define the sum of the series as the limit
if it exists and say that the series a converges unconditionally. Thus it is the limit of all finite partial sums. Because F is not totally ordered, and because there may be uncountably many finite partial sums, this is not a limit of a sequence of partial sums, but rather of a net.
Note, however thatneeds to be countable for the sum to be finite. To see this, suppose it is uncountable. Then some
would be uncountable, and we can estimate the sum
This definition is insensitive to the order of the summation, so the limit will not exist for conditionally convergent series. If, however, I is a well-ordered set (for example any ordinal), one may consider the limit of partial sums of the finite initial segments
If this limit exists, then the series converges. Unconditional convergence implies convergence, but not conversely, as in the case of real sequences. If X is a Banach space and I is well-ordered, then one may define the notion of absolute convergence. A series converges absolutely if
exists. If a sequence converges absolutely then it converges unconditionally, but the converse only holds in finite dimensional Banach spaces.
Note that in some cases if the series is valued in a space that is not separable, one should consider limits of nets of partial sums over subsets of I which are not finite.
Summations over arbitrary index sets
For real-valued series, an uncountable sum converges only if at most countably many terms are nonzero. Indeed, let
be the set of indices whose terms are greater than 1/n. Each In is finite (otherwise the series would diverge). The set of indices whose terms are nonzero is the union of the In by the Archimedean principle, and the union of countably many countable sets is countable by the axiom of choice.
Occasionally integrals of real functions are described as sums over the reals. The above result shows that this interpretation should not be taken too literally. On the other hand, any sum over the reals can be understood as an integral with respect to the counting measure, which accounts for the many similarities between the two constructions.
The proof goes forward in general first-countable topological vector spaces as well, such as Banach spaces; define In to be those indices whose terms are outside the n-th neighborhood of 0. Thus uncountable series can only be interesting if they are valued in spaces that are not first-countable.
Examples
Convergent series
Divergent series
Sequence transformations
Sequence
Thursday, March 20, 2008
In physics, mechanical energy describes the potential energy and kinetic energy present in the components of a mechanical system.
Scientists make simplifying assumptions to make calculations about how mechanical systems react. For example, instead of calculating the mechanical energy separately for each of the billions of molecules in a soccer ball, it is easier to treat the entire ball as one object. This means that only two numbers (one for kinetic mechanical energy, and one for potential mechanical energy) are needed for each dimension (for example, up/down, north/south, east/west) under consideration.
To calculate the energy of a system without any simplifying assumptions would require examining the state of all elementary particle(s) and considering all four fundamental interactions). This is usually only done for very small systems, such as those studied in particle physics.
Wednesday, March 19, 2008
Santiago de Tezanos (born June 25, 1971 in Montevideo, Uruguay) is a Uruguayan architect. He earned his degree at the School of Architecture, National University in Uruguay (Facultad de Arquitectura, Universidad de la República). He has an especialization in e-business and e-marketing, with 10+ years experience in these fields.
Since the start of his professional career, he has worked on a globalized scale, melding the worlds of architecture and information technology. This has led his practice to achieve a new level in the development of architectural design processes.
He is the principal of Santiago de Tezanos Architects, a de-localized architecture office with operations in Montevideo, Miami and Shanghai. The firm provides the full range of architectural services from simple consulting to complete projects. It has established collaborative relationships with several investors and contractor firms throughout the world, becoming their operations and design center. The firm's activities, based in the Business Process Outsourcing model, have turned it into a key player of this market.
The Harvard Graduate School of Design selected the firm as case study in Business Process Outsourcing in the field.
Recent projects (compeleted, under construction and in design phase) are scattered across the planet in such diverse locations as the United States, Costa Rica, United Kingdom, Ireland, Bermuda, Virgin Islands, New Zealand, Morocco and Cayman Islands.
For several years Mr. de Tezanos held a teaching position in "Architecture and Technology" at his college Alma Mater, Instituto Crandon, in Montevideo.
Some other of his interests and studies include film production, a strong background in information technology (being among the pioneer users of the Internet in Uruguay ca. 1991), lifelong activities as amateur astronomer and eclipse chaser, along with a command of several languages such as English, Spanish, French and German, together with basic knowledge of Greek.
Hotel Terre, Marrakech, Morocco
Maison des Iles, Sark, Channel Islands
Dixcart Bay Hotel expansion, Sark, Channel Islands
Oukaïmeden Ski Resort, Oukaïmeden, Morocco
Sanctuary Health&Spa Resort, Marrakech, Morocco
Olive Street Tower, Los Angeles, California, United States
FitzGerald House, Kilminnin, Ireland
Playa Potrero family residence, Playa Potrero, Costa Rica
Los Pargos family residence, Costa Rica
Residences in Morris County, New Jersey, United States
Private residence, Scottsdale, Arizona, United States
Private residence, St. Croix, US Virgin Islands
Landscape Architecture projects 100+ Residences, New Jersey, United States
Tuesday, March 18, 2008
Telephones - main lines in use: 17.336 million (1999)
Telephones - mobile cellular: 38.6 million (2004)
Telephone system: generally adequate, modern facilities
Radio broadcast stations: AM 208, FM 715, shortwave 1 (1998)
Radios: 13.1 million (1997)
Television broadcast stations: 228 (plus 2,112 repeaters); note - these figures include 11 television broadcast stations and 89 repeaters in the Canary Islands (September 1995)
Televisions: 16.2 million (1997)
Internet Service Providers (ISPs): 49 (1999)
See also: Broadband Internet access worldwide#Spain
Country code (Top-level domain): ES
Only recognised by Turkey.
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