Series Tests

Geometric Series Test

A series of the form
\sum_{n=0}^\infty ar^n

Geometric Series Test - Convergence

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

Geometric Series Test - Divergence

Diverges if |r| \geq 1

P-Series Test

A series of the form
\sum_{n=1}^\infty \frac{1}{n^p}

P-Series Test - Convergence

Converges if p > 1

P-Series Test - Divergence

Diverges if p \leq 1

Test for Divergence (Nth-Term Test)

For
\sum an , if \lim{n \to \infty} a_n \neq 0
or does not exist, the series diverges.

Test for Divergence Note

If
\lim{n \to \infty} an = 0
, the series may converge or diverge (use other tests).

Integral Test

For a positive, continuous, decreasing function f(n) such that an = f(n), \sum an
and
\int f(x)dx
either both converge or both diverge.

Comparison Test - Convergence

Compare
\sum an with a known series \sum bn
. If 0 \leq an \leq bn and
\sum bn converges, then \sum an
converges.

Comparison Test - Divergence

Compare
\sum an with a known series \sum bn
. If 0 \leq bn \leq an and
\sum bn diverges, then \sum an
diverges.

Limit Comparison Test

For
\sum an and \sum bn
, let
\lim{n \to \infty} \frac{an}{bn} = c , where c > 0. If \sum bn
converges,
\sum an also converges. If \sum bn
diverges,
\sum a_n
also diverges.

Alternating Series Test (Leibniz Test)

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

Ratio Test

For
\sum an , compute L = \lim{n \to \infty} \left|\frac{a{n+1}}{an}\right|
. 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
\sum an , compute L = \lim{n \to \infty} \sqrt[n]{|a_n|}
. 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
\sum |an| converges, then \sum an
converges absolutely.

Conditional Convergence

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