Integral Test

Integral Test: Let $f:[1,\infty)\to\mathbb{R}$ be positive, decreasing, and continuous, and set $a_n=f(n)$. Then $\sum_{n=1}^\infty a_n \text{ converges } \Longleftrightarrow \int_1^\infty f(x)\,dx \text{ converges}.$

The integral test connects series to calculus and provides error estimates by comparing partial sums to areas under $f$.