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

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

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

6. If    for some function    where

is easily evaluated, the Integral Test should be considered.

7. If the terms of the series are products involving powers of   

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.