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**friday616** Integral Test: Let $\displaystyle \sum$$\displaystyle a_{k}$ from k=1 to infinity be a series with positive terms and define a function f on [1,$\displaystyle \infty$) so that f(k)=$\displaystyle a_{k}$ for each positive integer k. Suppose that f is continuous and decreasing on [1,$\displaystyle \infty$) and that the limit of f(x) as x approaches infinity = 0. Then the series $\displaystyle \sum$$\displaystyle a_{k}$ from k=1 to infinity converges if and only if the improper integral $\displaystyle \int$ f from 1 to infinity converges. Furthermore, if S is the sum of the series and $\displaystyle s_{n}$ is the nth partial sum of the series, then 0 < S-$\displaystyle s_{n}$ < $\displaystyle \int$ f from n to infinity for each positive integer n.

Prove the integral test.