Strategies for Testing Series

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

$\sum {\frac {1}{n^{p}}}$  or  $\sum ar^{n},$ 

then the series is a  $p-$series or a geometric series

For the  $p-$series

$\sum {\frac {1}{n^{p}}},$ 

it is convergent if  $p>1$  and divergent if  $p\leq 1.$

For the geometric series

$\sum ar^{n},$ 

it is convergent if  $|r|<1$  and divergent if  $|r|\geq 1.$

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

then one of the comparison tests should be considered.

3. If you can see that

$\lim _{n\rightarrow \infty }a_{n}\neq 0,$ 

then you should use the Divergence Test or  $n$th term test.

4. If the series has the form

$\sum (-1)^{n}b_{n}$  or  $\sum (-1)^{n-1}b_{n}$ 

with  $b_{n}>0$  for all  $n,$  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  $n.$ 

For rational functions, you should use the Limit Comparison Test.

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

$\int _{a}^{\infty }f(x)~dx$ 

is easily evaluated, the Integral Test should be considered.

7. If all of the terms in the series have powers involving  $n,$ 

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.