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

  or   

then the series is a  series or a geometric series

For the  series

 

it is convergent if    and divergent if  

For the geometric series

 

it is convergent if    and divergent if  

2. If the series has a form similar to a  series or a geometric series, then one of the comparison tests should be considered.

3. If you can see that

 

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

4. If the series has the form

  or   

with    for all    then the Alternating Series Test should be considered.

5. If the series involves factorials or other products (including constants raised to the  th power), the Ratio Test should be considered.

NOTE: The Ratio Test should not be used for rational functions of  

6. If    for some function    where

 

is easily evaluated, the Integral Test should be considered (if all the hypothesis of the Integral Test are satisfied).

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.