Difference between revisions of "Convergence"
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:<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>. | + | 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. |
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| + | 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> | ||
Revision as of 03:22, December 6, 2008
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 (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 \Leftarrow} ),
- 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}a_{n}=0}
However there is not one unique convergence condition (if 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 \Rightarrow} )
Real series convergence
There are two common convergence, power series and alternating series. However there are many others.
Power Series convergence
A power 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}a_{n}=\sum^{\infty}_{n=1}b_{n}(x-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 \lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_{n}}|<1} .
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}} .