The Integral Test for Convergence of Infinite Series

无穷级数求和的积分审敛法 苏剑林 · 2009-08-12 · Mathematics series summation integral

This was a conjecture I came up with while studying series summation, and it has now turned out to be correct.

Given a series $\sum_{x=1}^{\infty} f(x)$, if

$\lim_{x -> \infty } \int f(x)dx -> \infty $, then the series diverges.

If $\lim_{x -> \infty } \int f(x)dx $ converges, then the series converges. more

For example:

For the series $\sum_{x=1}^{\infty} 1/x$, since $\int (1/x)dx=ln x$, $\lim_{x -> \infty}ln x -> \infty$, the series diverges.

For the series $\sum_{x=1}^{\infty} 1/{x^2}$, since $\int (1/{x^2})dx=-1/x$, $\lim_{x -> \infty}-1/x -> 0$, the series converges.

It turns out this result already existed; here I'm just stating it in a positive (affirmative) form.