Cauchy's Convergence Criterion for Series

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Theorem

A series $\displaystyle \sum_{i \mathop = 0}^\infty a_i$ is convergent if and only if for every $\epsilon > 0$ there is a number $N \in \N$ such that:

$\size {a_{n + 1} + a_{n + 2} + \cdots + a_m} < \epsilon$

holds for all $n \ge N$ and $m > n$.



Proof

Let:

$\displaystyle s_n = \sum_{i \mathop = 0}^n a_i$

Then $\sequence {s_n}$ is a sequence in $\R$.

From Cauchy's Convergence Criterion, $\sequence {s_n}$ is convergent if and only if it is a Cauchy sequence.

For $m > n$ we have:

$\size {s_m - s_n} = \size {a_{n + 1} + a_{n + 2} + \cdots + a_m}$

$\blacksquare$