# Strategies for Testing Series

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In general, there are no specific rules as to which test to apply to a given series.

Instead, we classify series by their form and give tips as to which tests should be considered.

This list is meant to serve as a guideline for which tests you should consider applying to a given series.

1. If the series is of the form

${\displaystyle \sum {\frac {1}{n^{p}}}}$  or  ${\displaystyle \sum ar^{n},}$
then the series is a  ${\displaystyle p-}$series or a geometric series
For the  ${\displaystyle p-}$series
${\displaystyle \sum {\frac {1}{n^{p}}},}$
it is convergent if  ${\displaystyle p>1}$  and divergent if  ${\displaystyle p\leq 1.}$
For the geometric series
${\displaystyle \sum ar^{n},}$
it is convergent if  ${\displaystyle |r|<1}$  and divergent if  ${\displaystyle |r|\geq 1.}$

2. If the series has a form similar to a  ${\displaystyle p-}$series or a geometric series,

then one of the comparison tests should be considered.

3. If you can see that

${\displaystyle \lim _{n\rightarrow \infty }a_{n}\neq 0,}$
then you should use the Divergence Test or  ${\displaystyle n}$th term test.

4. If the series has the form

${\displaystyle \sum (-1)^{n}b_{n}}$  or  ${\displaystyle \sum (-1)^{n-1}b_{n}}$
with  ${\displaystyle b_{n}>0}$  for all  ${\displaystyle n,}$  then the Alternating Series Test should be considered.

5. If the series involves factorials or other products, the Ratio Test should be considered.

NOTE: The Ratio Test should not be used for rational functions of  ${\displaystyle n.}$
For rational functions, you should use the Limit Comparison Test.

6. If  ${\displaystyle a_{n}=f(n)}$  for some function  ${\displaystyle f(x)}$  where

${\displaystyle \int _{a}^{\infty }f(x)~dx}$
is easily evaluated, the Integral Test should be considered.

7. If the terms of the series are products involving powers of  ${\displaystyle n,}$

then the Root Test should be considered.

NOTE: These strategies are used for determining whether a series converges or diverges.

However, these are not the strategies one should use if we are determining whether or not a

series is absolutely convergent.