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Cauchy criterion for convergence
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(Theorem)
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A series $\sum_{i=0}^\infty a_i$ in a Banach space $(V,\|\cdot\|)$ is convergent iff for every $\varepsilon>0$ there is a number $N\in\mathbb{N}$ such that $$\|a_{n+1}+a_{n+2}+\cdots+a_{n+p}\|<\varepsilon$$ holds for all $n>N$ and $p\geq1$
First define $$s_n:=\sum_{i=0}^n a_i.$$ Now, since $V$ is complete, $(s_n)$ converges if and only if it is a Cauchy sequence, so if for every $\varepsilon>0$ there is a number $N$ such that for all $n,m>N$ holds: $$\|s_m-s_n\|<\varepsilon.$$ We can assume $m>n$ and thus set $m=n+p$ The series is convergent iff $$\|s_{n+p}-s_n\|=\|a_{n+1}+a_{n+2}+\cdots+a_{n+p}\|<\varepsilon.$$
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Cross-references: Cauchy sequence, converges, complete, number, iff, Banach space, series
There are 5 references to this entry.
This is version 11 of Cauchy criterion for convergence, born on 2003-01-16, modified 2007-12-15.
Object id is 3894, canonical name is CauchyCriterionForConvergence.
Accessed 14629 times total.
Classification:
| AMS MSC: | 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences) |
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Pending Errata and Addenda
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