Integral Test

$\sum a_n$ and $\int_1^{\infty}f(x)\,dx$ converge or diverge TOGETHER, when $a_n=f(n)$ for positive, decreasing, continuous $f$ .

The Integral Test says: if $f$ is positive, continuous, and decreasing on $[1,\infty)$ with $f(n)=a_n$ , then the series $\sum_{n=1}^{\infty}a_n$ and the improper integral $\int_1^{\infty}f(x)\,dx$ CONVERGE OR DIVERGE TOGETHER — the integral's area both bounds and is bounded by the series' terms, since a decreasing function's rectangles and the curve stay within a fixed distance of each other forever.

Examples

$\sum\frac{1}{n^2}$ converges by the Integral Test, since $\int_1^{\infty}\frac{1}{x^2}\,dx=1$ converges.