Series Tests

Geometric Series Test

A series of the form $<br /> \sum_{n=0}^\infty ar^n<br />$

Geometric Series Test - Convergence

Converges if |r| < 1. The sum is $<br /> \frac{a}{1 - r}<br />$

Geometric Series Test - Divergence

Diverges if $|r| \geq 1$

P-Series Test

A series of the form $<br /> \sum_{n=1}^\infty \frac{1}{n^p}<br />$

P-Series Test - Convergence

Converges if p > 1

P-Series Test - Divergence

Diverges if $p \leq 1$

Test for Divergence (Nth-Term Test)

For $<br /> \sum a<em>n$, if $\lim</em>{n \to \infty} a_n \neq 0<br />$ or does not exist, the series diverges.

Test for Divergence Note

If $<br /> \lim<em>{n \to \infty} a</em>n = 0<br />$, the series may converge or diverge (use other tests).

Integral Test

For a positive, continuous, decreasing function $f(n)$ such that $a<em>n = f(n)$, $\sum a</em>n<br />$ and $<br /> \int f(x)dx<br />$ either both converge or both diverge.

Comparison Test - Convergence

Compare $<br /> \sum a<em>n$ with a known series $\sum b</em>n<br />$. If $0 \leq a<em>n \leq b</em>n$ and $<br /> \sum b<em>n$ converges, then $\sum a</em>n<br />$ converges.

Comparison Test - Divergence

Compare $<br /> \sum a<em>n$ with a known series $\sum b</em>n<br />$. If $0 \leq b<em>n \leq a</em>n$ and $<br /> \sum b<em>n$ diverges, then $\sum a</em>n<br />$ diverges.

Limit Comparison Test

For $<br /> \sum a<em>n$ and $\sum b</em>n<br />$, let $<br /> \lim<em>{n \to \infty} \frac{a</em>n}{b<em>n} = c$, where $c > 0$. If $\sum b</em>n<br />$ converges, $<br /> \sum a<em>n$ also converges. If $\sum b</em>n<br />$ diverges, $<br /> \sum a_n<br />$ also diverges.

Alternating Series Test (Leibniz Test)

For an alternating series $<br /> \sum (-1)^n a<em>n$, where an > 0, converges if $a<em>n$ is decreasing and $\lim</em>{n \to \infty} a_n = 0<br />$

Ratio Test

For $<br /> \sum a<em>n$, compute $L = \lim</em>{n \to \infty} \left|\frac{a<em>{n+1}}{a</em>n}\right|<br />$. If $L < 1$, the series converges absolutely. If $L > 1$ or $L = \infty$, the series diverges. If $L = 1$, the test is inconclusive.

Root Test

For $<br /> \sum a<em>n$, compute $L = \lim</em>{n \to \infty} \sqrt[n]{|a_n|}<br />$. If $L < 1$, the series converges absolutely. If $L > 1$ or $L = \infty$, the series diverges. If $L = 1$, the test is inconclusive.

Absolute Convergence Test

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

Conditional Convergence

A series $<br /> \sum a<em>n$ converges conditionally if $\sum a</em>n<br />$ converges, but $<br /> \sum |a_n|<br />$ diverges.