MathAdmin: Created page with "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 cons..."

2017-10-30T18:48:00Z

Created page with "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 cons..."

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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

:::<math style="vertical-align: -10px">\sum \frac{1}{n^p} </math>  or  <math style="vertical-align: -5px">\sum ar^n,</math> 

:then the series is a  <math style="vertical-align: -4px">p-</math>series or a geometric series

:For the  <math style="vertical-align: -4px">p-</math>series

:::<math>\sum \frac{1}{n^p},</math> 

:it is convergent if  <math style="vertical-align: -4px">p>1</math>  and divergent if  <math style="vertical-align: -4px">p\le 1.</math>

:For the geometric series

:::<math>\sum ar^n,</math> 

:it is convergent if  <math style="vertical-align: -5px">|r|<1</math>  and divergent if  <math style="vertical-align: -4px">|r|\ge 1.</math>

'''2.''' If the series has a form similar to a  <math style="vertical-align: -4px">p-</math>series or a geometric series,

:then one of the comparison tests should be considered.

'''3.''' If you can see that

:::<math>\lim_{n\rightarrow \infty} a_n \neq 0,</math> 

:then you should use the Divergence Test or  <math style="vertical-align: 0px">n</math>th term test.

'''4.''' If the series has the form

:::<math style="vertical-align: -6px">\sum (-1)^n b_n</math>  or  <math style="vertical-align: -6px">\sum (-1)^{n-1} b_n</math> 

:with  <math style="vertical-align: -4px">b_n>0</math>  for all  <math style="vertical-align: -4px">n,</math>  then the Alternating Series Test should be considered.

'''5.''' If the series involves factorials or other products, the Ratio Test should be considered.

:<u>NOTE:</u> The Ratio Test should not be used for rational functions of  <math style="vertical-align: 0px">n.</math> 

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

'''6.''' If  <math style="vertical-align: -5px">a_n=f(n)</math>  for some function  <math style="vertical-align: -5px">f(x)</math>  where

:::<math>\int_a^\infty f(x)~dx</math> 

:is easily evaluated, the Integral Test should be considered.

'''7.''' If the terms of the series are products involving powers of  <math style="vertical-align: -4px">n,</math> 

:then the Root Test should be considered.

<u>NOTE:</u> 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.

Strategies for Testing Series - Revision history

MathAdmin: Created page with "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 cons..."