Series Rules

Divergence Test

If $\lim<em>{n\to\infty} a</em>n \ne 0$, then the series $\sum a_n$ Diverges. If it equals 0, the test is inconclusive.

Geometric Series

A series of the form $\sum_{n=1}^\infty a(r)^{n-1}$ converges if |r| < 1 and diverges if $|r| \ge 1$.

P-Series Test

For a series of the form $\sum_{n=1}^\infty \frac{1}{n^p}$ , it converges if p > 1 and diverges if $p \le 1$.

Telescoping Series

A series where intermediate terms cancel out. It converges if its partial sum $\lim<em>{n\to\infty} S</em>n$ approaches a finite value; otherwise, it diverges.

Integral Test

If $f(x)$ is positive, continuous, and decreasing for $x \ge n$, then $\sum a<em>n$ converges if $\int</em>n^\infty f(x)dx$ converges (is finite) and diverges if the integral diverges (is $\pm\infty$ or DNE).

Ratio Test

For $\lim<em>{n\to\infty} \left| \frac{a</em>{n+1}}{a_n} \right|$: if $< 1$, converges; if $> 1$ or $+\infty$, diverges; if $= 1$, inconclusive.

Root Test

For $\lim<em>{n\to\infty} \sqrt[n]{|a</em>n|}$: if $< 1$, converges; if $> 1$, diverges; if $= 1$, inconclusive.

Direct Comparison Test

Given $0 \le a<em>n \le b</em>n$: if $\sum b<em>n$ converges, then $\sum a</em>n$ converges; if $\sum a<em>n$ diverges, then $\sum b</em>n$ diverges.

Limit Comparison Test

If $\lim<em>{n\to\infty} \frac{a</em>n}{b<em>n} = L$ where $L$ is finite and positive, then $\sum a</em>n$ and $\sum b_n$ either both converge or both diverge.

Alternating Series Test

An alternating series $\sum (-1)^n a<em>n$ converges if $a</em>n$ is positive, decreasing, and $\lim<em>{n\to\infty} a</em>n = 0$.

Absolute Value Test

If $\sum |a<em>n|$ converges, then $\sum a</em>n$ converges (absolutely convergent).