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Integral Test

Let $\sum u_k$ be a series with Positive terms and let $f(x)$ be the function that results when $k$ is replaced by $x$ in the Formula for $u_k$. If $f$ is decreasing and continuous for $x \geq 1$ and

\lim_{x\to \infty} f(x) = 0,


\sum_{k=1}^\infty u_k


\int^\infty_t f(x)\,dx

both converge or diverge, where $1 \leq t <\infty$. The test is also called the Cauchy Integral Test or Maclaurin Integral Test.

See also Convergence Tests


Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 283-284, 1985.

© 1996-9 Eric W. Weisstein