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 just 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 (including constants raised to the  $n$th power), the Ratio Test should be considered. NOTE: The Ratio Test should not be used for rational functions of  $n.$

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 (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.