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

$\sum {\frac {1}{n^{p}}}$ 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 the terms of the series are products involving powers of  $n,$ 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.