Proof of the Week for 20120813

Cauchy convergence criterion for a series of real numbers: The series \sum_{n=1}^\infty a_n converges if and only if for \epsilon > 0 , there exists N \in \mathbb{N} such that whenever n, m \ge N , \left|a_{m+1} + a_{m+2} + \cdots + a_n\right| < \epsilon .

Proof: Let \left(S_n\right) and \left(S_m\right) be sequences of partial sums of the series \sum a_n , such that S_n = \sum_{p=1}^n a_p , S_m = \sum_{q=1}^m a_q , where n > m . Invoking the Cauchy convergence criterion for sequences, we have \left|S_n - S_m\right| < \epsilon . \left|S_n - S_m\right| = \left|a_{m+1} + a_{m+2} + \cdots + a_n\right| < \epsilon .

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