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The Integral Test

The Integral Test for convergence is a method used to test convergence of an infinite series of nonnegative terms.

The series

\begin{displaymath}\sum_{n=1}^{\infty} a_n \end{displaymath}

converges if and only if the integral

\begin{displaymath}\int_1^{\infty} f(x)   dx \end{displaymath}

is finite, where $f(x)$ is a positive, non-increasing and continuous function defined on the interval $[1,\infty)$ and $f(n) = a_n$ an for all $n$.

Dina J. Solitro-Rassias