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 $f$.

Proof sketch (optional): Compare the area under $f$ on $[n,n+1]$ with rectangles of height $f(n)$ and $f(n+1)$ to trap the tail $\sum_{k=n}^\infty f(k)$ between integrals.