# List of Series Tests

The series of interest will always by symbolized as the sum,
as *n* goes from 1 to infinity, of *a*[*n*]. In addition,
any auxilliary sequence
will be symbolized as the sum, as *n* goes from 1 to infinity, of
*b*[*n*]. Or, symbolically,
and
.

Click on the name of the test to get more information on the test.

## The Common Series Tests

If the limit of *a*[*n*] is not zero, or does not exist,
then
the sum diverges.

If you can define *f* so that it is a continuous, positive,
decreasing
function from 1 to infinity (including 1) such that
*a*[*n*]=*f(n)*, then the sum
will converge if and only if the integral of *f* from 1 to infinity
converges.

Please note that this does *not* mean that the sum of the series
is that same as the value of the integral. In most cases, the two will be
quite different.

Let *b*[*n*] be a second series. Require that all
*a*[*n*] and *b*[*n*] are positive. If
*b*[*n*] converges, and
*a*[*n*]<=*b*[*n*] for all *n*, then
*a*[*n*] also converges. If
the sum of *b*[*n*] diverges, and
*a*[*n*]>=*b*[*n*] for all *n*, then the sum
of *a*[*n*] also
diverges.

Let *b*[*n*] be a second series. Require that all
*a*[*n*] and *b*[*n*] are positive.

- If the limit of
*a*[*n*]/*b*[*n*] is positive,
then the sum of *a*[*n*] converges
if and only if the sum of *b*[*n*] converges.
- If the limit of
*a*[*n*]/*b*[*n*] is zero, and the
sum of *b*[*n*] converges, then the
sum of *a*[*n*] also converges.
- If the limit of
*a*[*n*]/*b*[*n*] is infinite, and
the sum of *b*[*n*] diverges, then the
sum of *a*[*n*] also diverges.

If
*a*[*n*]=(-1)^(*n*+1)*b*[*n*], where
*b*[*n*] is positive, decreasing, and converging to
zero, then the sum of *a*[*n*] converges.

If the sum of |*a*[*n*]| converges, then the sum of
*a*[*n*] converges.

If the limit of |*a*[*n*+1]/*a*[*n*]| is less
than
1, then the series (absolutely)
converges. If the limit is larger than one, or infinite, then the series
diverges.

If the limit of |*a*[*n*]|^(1/*n*) is less than one,
then
the series (absolutely)
converges. If the limit is larger than one, or infinite, then the series
diverges.

Copyright © 1996 Department
of Mathematics, Oregon State University
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