P-Series and Ratio Tests


Recall that the geometric series is a series where the general term is a constant multiple of rn where r is a constant and n is the index of summation.  We we will look at exponents where the constant is the power and the base is the index.

     Definition of a P-Series

If p is a real number then the series 


is called a P-Series.




is a P-series with p  =  -1/2.  Recall that roots are just 1/2 powers and pulling up from the denominator is just changing the sign of the exponent.


is a P-series with p  =  1.  This is just the harmonic series.


is not a P-series since the index is in the exponent.  In fact the above series is a geometric series with r  =  1/2.


P-Series Test

The following test tells us when the p-series converges.

        Theorem:  P-Series Test
Consider the series


  1. If p > 1 then the series converges

  2. If 0 < p < 1 then the series diverges



The harmonic series


 diverges by the P-series test since 

        p  =  1  <  1


The series 


diverges since 

        p  =  2  > 1

Ratio Test

If a series converges then the terms must approach zero as n gets large.  However, we have seen that even if the terms approach zero, the series may still diverge.  For example, the harmonic series diverges, however the terms do not approach zero fast enough for convergence.  The next theorem gives us a method that will often detect whether the terms approach zero fast enough.

             Theorem:  The Ratio Test

 Let  be a series with nonzero terms.  Then

  1. If 


    then the series converges.

  2. If   


    then the series diverges.

  3. If  


    then try another test.


The ratio test is especially useful when powers or factorials appear in the general term.  For powers, we have

                   =  b

For factorials, note for example

        7!             (7)(6)(5)(4)(3)(2)    
                =                                      =      7              
        6!               (6)(5)(4)(3)(2)

In general,

                       =  n+1             



Determine the convergence or divergence of


We use the Ratio Test:


Hence the series converges by the Ratio Test


Determine the convergence or divergence of


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