Convergence Tests Decision Flowchart with Examples Cheat Sheet

Key Facts

• The divergence test says if $\lim_{n \to \infty} a_n \ne 0$ or the limit does not exist, then $\sum a_n$ diverges.
• A geometric series $\sum_{n=0}^{\infty} ar^n$ converges when $|r| < 1$ and has sum $\frac{a}{1-r}$.
• A p-series $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges when $p > 1$ and diverges when $p \le 1$.
• The integral test applies when $a_n = f(n)$ and $f(x)$ is positive, continuous, and decreasing, and $\sum a_n$ converges exactly when $\int_1^{\infty} f(x)\,dx$ converges.
• The direct comparison test says if $0 \le a_n \le b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.
• The limit comparison test says if $a_n > 0$, $b_n > 0$, and $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ with $0 < c < \infty$, then $\sum a_n$ and $\sum b_n$ have the same behavior.
• The ratio test uses $L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$: the series converges if $L < 1$, diverges if $L > 1$, and is inconclusive if $L = 1$.
• The alternating series test says $\sum (-1)^n b_n$ converges if $b_n \ge 0$, $b_{n+1} \le b_n$, and $\lim_{n \to \infty} b_n = 0$.

Vocabulary

Infinite series

An infinite series is a sum of infinitely many terms, written as $\sum_{n=1}^{\infty} a_n$.

Convergence

A series converges if its sequence of partial sums approaches a finite number.

Divergence

A series diverges if its partial sums do not approach a finite number.

Absolute convergence

A series $\sum a_n$ converges absolutely if $\sum |a_n|$ converges.

Conditional convergence

A series $\sum a_n$ converges conditionally if $\sum a_n$ converges but $\sum |a_n|$ diverges.

Partial sum

The $N$th partial sum is $S_N = \sum_{n=1}^{N} a_n$, the sum of the first $N$ terms.