Integral Test and p-Series

The Integral Test

Consider a series S a_n such that 

        a_n > 0     and     a_n > a_n+1  

We can plot the points (n,a_n) on a graph and construct rectangles whose bases are of length 1 and whose heights are of length a_n.  If we can find a continuous function f(x) such that 

        f(n)  =  a_n

       
then notice that the area of these rectangles (light blue plus purple) is an upper Reimann sum for the area under the graph of f(x).  Hence 

       

Similarly, if we investigate the lower Reimann sum we see that

       
Since the first term is of a series has no bearing in its convergence or divergence, this proves the following theorem:

Theorem:  The Integral Test Let f(x) be a positive continuous function that is eventually  decreasing and let f(n) = an Then             converges if and only if               converges.

 Note the contrapositive:

Corollary                 diverges if and only if                diverges.

Example:

1. Consider the series:  
   
          

   We use the Integral Test:

   Let 
   
           f(x) = 1/x²  

            

   Hence by the Integral Test 
   
            
   
   converges.

   What does it converge to?  We use a calculator to get 1.635...
   
   

2. Consider the series 
   
          

   We use the Integral Test.  Let 
   
            

            
   
   Hence by the Integral Test 
   
          
   
   diverges.

P-Series Test

A special case of the integral test is when  

                     1
        a_n  =            
                    n^p 

for some p.  The theorem below discusses this.

Theorem:  P-Series Test Consider the series                     If p > 1 then the series converges If 0 < p < 1 then the series diverges

Proof:

We use the integral test with the function 

                        1
        f(x)  =            
                       x^p 

For p not equal to 1, 

       

Note that this limit converges if

        -p + 1 < 0 

or 

        p > 1

The limit diverges for p < 1

For p = 1 we have the harmonic series

       

and the integral test gives:

Another proof that the harmonic series diverges.

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