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

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

then

$$\sum_{k=1}^\infty u_k$$

and

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

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

References

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