Series

A series ${\sum }_{n=1}^{\infty }{a}_n$ converges exactly when its partial sums $S_N={\sum }_{n=1}^N{a}_n$ approach a finite limit as $N\to \infty$. A series with positive terms converges if its terms are eventually bounded above by the terms of a known convergent series; this is the comparison test. In particular, $\sum 1/n^p$ converges for $p>1$ and diverges for $p\le 1$. To find a radius of convergence $R$ for a power series $\sum c_n{x}^n$, use the ratio test: $R={\lim }_{n\to \infty }|c_n/c_{n+1}|$ when this limit exists. For series with positive terms, the integral test compares $\sum a_n$ to ${\int }_1^{\infty }f(x)dx$ when $a_n=f(n)$ for a positive decreasing function $f$.