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Expanded the page, adding more examples of tests
:<math>\lim_{N\rightarrow\infty}\sum^{N}_{n=1}a_{n}=L</math>
:<math>\lim_{N\rightarrow\infty}\sum^{N}_{n=1}a_{n}=L</math>
for <math>L</math> a finite number, <math>|L|<\infty</math>. The terms in the limit are known as [[partial sum]]s. If the partial sums approach <math>\infty</math> or <math>-\infty</math>, the series is said to '''diverge'''. If the partial sums do not approach a single value, finite or infinite, then the series is said to have no limit.
for <math>L</math> a finite number, <math>|L|<\infty</math>.<ref>[http://mathworld.wolfram.com/ConvergentSeries.html Convergent Series] from mathworld.wolfram.com</ref> The terms in the limit are known as partial sums. If the partial sums approach <math>\infty</math> or <math>-\infty</math>, the series is said to '''diverge'''. If the partial sums do not approach a single value, finite or infinite, then the series is said to have no limit.
Convergence of the series will occur only if (<math>\Leftarrow</math>),
Convergence of the series will occur only if (<math>\Leftarrow</math>),
:<math>\lim_{n\rightarrow\infty}a_{n}=0</math>
:<math>\lim_{n\rightarrow\infty}a_{n}=0</math>
However, there is not one unique convergence condition (if or <math>\Rightarrow</math>)
However, there is not one unique convergence condition (if or <math>\Rightarrow</math>).
==Real series convergence==
==Types of Convergence==
A series is said to converge [[Absolute value|absolutely]] if the limit
===Power Series convergence===
:<math>
\lim_{N\rightarrow\infty}\sum^{N}_{n=1} \left| a_{n} \right|=L
</math>
If this limit holds for <math>a_n</math> but not for <math>|a_n|</math>, then the series is said to converge conditionally. If a series converges absolutely, then the series, <math>\sum^{N}_{n=1} a_{n}</math>, must also converge.
==Real series convergence==
There are many test for checking whether a series converges. A particularly important class of series are power series which have the form,
:<math>\sum^{\infty}_{n=1}a_{n}=\sum^{\infty}_{n=1}b_{n}(x-a)^{n}</math>
===D'Alembert's Ratio Test===
D'Alembert's ratio test states that for the limit:
:<math>
L = \lim_{n\rightarrow\infty} \left| \frac{a_{n+1}}{a_{n}} \right|
</math>
the series will converge if L<1.<ref name=MathematicalMethods>K.F. Riley, M.P. Hobson, S.J. Bence, ''Mathematical Methods for Physics and Engineering'', Cambridge University Press, 3<sup>rd</sup> ed., 2006</ref> if L>1 the series diverges and if L=0 the test is inconclusive and an alternative test must be used.
===Alternating Series convergence===
===Alternating Series convergence===
This will converge if <math>0\leq a_{n+1}<a_{n}</math>.
This will converge if <math>0\leq a_{n+1}<a_{n}</math>.
===Root test===
===Cauchy's Root test===
Cauchy's root test states that the limit:
:<math>
L = \lim_{N\rightarrow\infty}\sum^{N}_{n=1} \sqrt[n]{a_{n}}
</math>
can be used to test convergence.<ref>[http://tutorial.math.lamar.edu/Classes/CalcII/RootTest.aspx Cauchy's root test] from tutorial.math.lamar.edu</ref> If:
*L<1, then the series converges absolutely and so must converge
*L>1, then the series diverges
*L=0, then the series may converge or diverge, another test must be used.
===Limit Comparison Test===
The limit comparison test allows one to test one series, <math>\Sigma a_n</math>, for convergence by comparing it to another series, <math>\Sigma b_n</math>, that is known to converge or diverge.<ref>[http://mathworld.wolfram.com/LimitComparisonTest.html Limit comparison test] from mathworld.wolfram.com</ref> This test is only valid if all terms in the two series are positive. It states that if the limit
:<math>
\rho = \lim_{n\rightarrow \infty} \frac{a_n}{b_n}
</math>
is finite and <math>\rho>0</math> then <math>\Sigma a_n</math> converges if <math>\Sigma b_n</math> converges or diverges if <math>\Sigma b_n</math> diverges.
===Integral Test===
Suppose we have a [[function]], <math>f(x)</math> for which <math>f(n)=a_n</math>. Then if the limit:
:<math>
L = \lim_{N\rightarrow \infty} \int^N f(x) \, dx
</math>
exists, then the series <math>\Sigma a_n</math> must converge. If the [[integral]] cannot be evaluated, then the series diverges.<ref>[http://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx Integral test] from tutorial.math.lamar.edu</ref> This forms a very powerful technique for testing convergence. Furthermore, if the limit does exist, then upper and lower bounds for the series can be found:
:<math>
L \leq \sum^{N}_{n=n_0}a_{n} \leq L + f(n_0)
</math>
where n<sub>0</sub> is the starting point of the series. As an example, consider the harmonic series, <math>\Sigma^{\infty}_{n=1} n^{-1}</math>. Evaluating integral using <math>f(x)=x^{-1}</math> we get:
==Complex series convergence==
:<math>
\lim_{N\rightarrow \infty} \int^N \frac{1}{x} \, dx = \lim_{N\rightarrow \infty} \left( \ln{N} - \ln{1} \right)
</math>
The first [[natural logarithm]] goes to infinity as N goes to infinity. So although each term get smaller and smaller, the series diverges.
==Intervals of convergence==
==Intervals of convergence==
The interval of convergence, also known as the [[radius of convergence]], describes the range of values for which an [[infinite series]] converges.<ref>[http://mathworld.wolfram.com/RadiusofConvergence.html Radius of convergence] from mathworld.wolfram.com</ref> For [[real number|real]] series this is an interval or region on the number line and can be expressed in the form a<x<b. As [[complex numbers]] have two components, this interval is transforms from a 1 dimensional line to the 2 dimensional area of a circle. Depending on the series, the interval of convergence may include some numbers, all numbers or no numbers. For example, the [[exponential]] function can be written as a power series:
:<math>
e^x = \sum^{\infty}_{n=0} \frac{x^n}{n!}
</math>
This series converges for all x, real or complex. The [[binomial expansion]] of the function (1+x)<sup>-1</sup> is:
:<math>
\frac{1}{1+x} = \sum^{\infty}_{n=0}x^n
</math>
and only converges for |x|<1.
==References==
{{reflist}}
==See also==
*[[Series (mathematics)]]
[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Calculus]]