imported>Daniel Mietchen |
imported>Chunbum Park |
| Line 1: |
Line 1: |
| The '''[[eurozone crisis]]''' that started in 2010 arose from doubts about the ability of some eurozone governments to service their debts. .
| | A '''[[geometric series]]''' is a [[series (mathematics)|series]] associated with a [[geometric sequence]], |
| The financial assistance given to those governments has failed restore the confidence of the markets, and bond market investors have become reluctant to buy the bonds being issued by some other eurozone governments. There are uncertainties about the willingness of the major eurozone governments to provide the further assistance that may be needed, and fears that there may be a breakup of the eurozone and a global financial crisis if they do not.
| | i.e., the ratio (or quotient) ''q'' of two consecutive terms is the same for each pair. |
|
| |
|
| ====Overview====
| | Thus, every geometric series has the form |
| The crisis started early in 2010 with the revelation that, without external assistance, the Greek government would be forced to default on its debt. The assistance that was provided by other eurozone governments enabled the Greek government to continue to roll-over maturing debts until, in the latter half of 20ll, it became evident that a default could no longer be avoided. In the meantime, investors' fears of default had increased the cost of borrowing to other eurozone governments, making it necessary to provide financial assistance to the governments of both Ireland and Portugal. By September 2011, the international community had become aware of the danger that a Greek government default, and that its repercussions could administer a shock to the world economy comparable to the shock that triggered the Great Recession. Plans were initiated to provide the financial support needed to avoid a comparable malfunction of the global financial system. Substantial political obstacles would have to be overcome before such plans could be put into effect.
| | :<math> |
| | a + aq + aq^2 + aq^3 + \cdots |
| | </math> |
| | where the quotient (ratio) of the (''n''+1)th and the ''n''th term is |
| | :<math> |
| | \frac{a q^{n}}{aq^{n-1}} = q. |
| | </math> |
|
| |
|
| ====Background to the crisis====
| | The sum of the first ''n'' terms of a geometric sequence is called the ''n''-th partial sum (of the series); its formula is given below (''S''<sub>''n''</sub>). |
| In 1991, leaders of the 15 countries that then made up the European Union, set up a monetary union with a single currency. There were strict criteria for joining (including targets for inflation, interest rates and budget deficits), and other rules that were intended to preserve its members' fiscal sustainability were added later. No provision was made for the expulsion of countries that did not comply with its rules, nor for the voluntary departure of those who no longer wished to remain, but it was intended to impose financial penalties for breaches.
| |
|
| |
|
| Greece joined, what by then was known as the eurozone, in 2001, Slovenia in 2007, Cyprus and Malta in 2008, Slovakia in 2009.
| | An infinite geometric series (i.e., a series with an infinite number of terms) converges if and only if |''q''|<1, in which case its sum is <math> a \over 1-q </math>, where ''a'' is the first term of the series. |
| The current membership comprises Belgium, Germany¸ Ireland, Greece, Spain, France, Italy, Cyprus, Luxembourg, Malta, The Netherlands, Austria, Portugal, Slovenia, Slovakia, and Finland. Bulgaria, Czech Republic.
| |
|
| |
|
| The non-members of the eurozone among members of the European Union are Denmark, Estonia, Latvia, Lithuania, Hungary, Poland, Romania, Sweden and the United Kingdom.
| | In finance, since compound [[interest rate|interest]] generates a geometric sequence, |
| | regular payments together with compound interest lead to a geometric series. |
|
| |
|
| ''[[Eurozone crisis|.... (read more)]]'' | | ''[[Geometric series|.... (read more)]]'' |
| {|align="center" cellpadding="5" style="background:lightgray; width:95%; border: 1px solid #aaa; margin:10px; font-size: 92%;"
| |
| | In addition to the above text, this article comprises:<br> - a [[Eurozone crisis/Addendum#Crisis development by country|'''country-by-country summary''']] of the development of the crisis;<br> - [[Eurozone crisis/Timelines|'''links to contemporary reports''']] of the main events of the crisis;<br> - notes on [[Eurozone crisis/Tutorials#The debt trap|'''the debt trap''']], the eurozone's departures from [[Eurozone crisis/Tutorials#Departures from optimum currency area criteria| '''optimum currency area criteria''']], and on [[Eurozone crisis/Tutorials#The eurobond proposal|'''the eurobond proposal''']]; and,<br> - tabulations of the [[Eurozone crisis/Addendum#Fiscal characteristics|'''fiscal characteristics of the PIIGS countries''']] , and their [[Eurozone crisis/Addendum#GDP growth|'''GDP growth rates''']]
| |
| |}
| |
A geometric series is a series associated with a geometric sequence,
i.e., the ratio (or quotient) q of two consecutive terms is the same for each pair.
Thus, every geometric series has the form
where the quotient (ratio) of the (n+1)th and the nth term is
The sum of the first n terms of a geometric sequence is called the n-th partial sum (of the series); its formula is given below (Sn).
An infinite geometric series (i.e., a series with an infinite number of terms) converges if and only if |q|<1, in which case its sum is 
, where a is the first term of the series.
In finance, since compound interest generates a geometric sequence,
regular payments together with compound interest lead to a geometric series.
.... (read more)