Convergence

In mathematics convergence of an infinite series occurs when,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N\rightarrow\infty}\sum^{N}_{n=1}a_{n}=L}

for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} a finite number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |L|<\infty} . The terms in the limit are known as partial sums. If the partial sums approach Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty} , 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 ($\Leftarrow$),

$\lim _{n\rightarrow \infty }a_{n}=0$

However, there is not one unique convergence condition (if or $\Rightarrow$).

Types of Convergence

A series is said to converge absolutely if the limit

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N\rightarrow\infty}\sum^{N}_{n=1} \left| a_{n} \right|=L }

If this limit holds for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n} but not for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |a_n|} , then the series is said to converge conditionally. If a series converges absolutely, then the series, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum^{N}_{n=1} a_{n}} , 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,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum^{\infty}_{n=1}a_{n}=\sum^{\infty}_{n=1}b_{n}(x-a)^{n}}

D'Alembert's Ratio Test

D'Alembert's ratio test states that for the limit:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \lim_{n\rightarrow\infty} \left| \frac{a_{n+1}}{a_{n}} \right| }

the series will converge if L<1. if L>1 the series diverges and if L=0 the test is inconclusive and an alternative test must be used.

Alternating Series convergence

An alternating series is of the form,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum^{\infty}_{n=1}(-1)^{n}a_{n}}

This will converge if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq a_{n+1}<a_{n}} .

Cauchy's Root test

Cauchy's root test states that the limit:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \lim_{N\rightarrow\infty}\sum^{N}_{n=1} \sqrt[n]{a_{n}} }

can be used to test convergence. 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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma a_n} , for convergence by comparing it to another series, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma b_n} , that is known to converge or diverge. This test is only valid if all terms in the two series are positive. It states that if the limit

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = \lim_{n\rightarrow \infty} \frac{a_n}{b_n} }

is finite and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho>0} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma a_n} converges if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma b_n} converges or diverges if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma b_n} diverges.

Integral Test

Suppose we have a function, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} for which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(n)=a_n} . Then if the limit:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \lim_{N\rightarrow \infty} \int^N f(x) \, dx }

exists, then the series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma a_n} must converge. If the integral cannot be evaluated, then the series diverges. 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L \leq \sum^{N}_{n=n_0}a_{n} \leq L + f(n_0) }

where n₀ is the starting point of the series. As an example, consider the harmonic series, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma^{\infty}_{n=1} n^{-1}} . Evaluating integral using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^{-1}} we get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{N\rightarrow \infty} \int^N \frac{1}{x} \, dx = \lim_{N\rightarrow \infty} \left( \ln{N} - \ln{1} \right) }

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

The interval of convergence, also known as the radius of convergence, describes the range of values for which an infinite series converges. For 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^x = \sum^{\infty}_{n=0} \frac{x^n}{n!} }

This series converges for all x, real or complex. The binomial expansion of the function (1+x)^-1 is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{1+x} = \sum^{\infty}_{n=0}x^n }

and only converges for |x|<1.

See also

• Series (mathematics)