Root and Ratio Test

The Ratio Test

Theorem:  The Ratio Test  Let be a series then If            then the series converges absolutely. If              then the series diverges. If               then try another test.

Proof:  Suppose that  


         

then for the tail,

        |a_n+1|  <  R |a_n|

        |a_n+2|  <  R |a_n+1|  <  R² |a_n|  ...

        |a_n+k|  <  R^k |a_n|  

So that

       

Which converges by the GST.

Example

Determine the convergence or divergence of

       

We use the Ratio Test:


           

Hence the series converges by the Ratio Test

Exercises  

Determine the convergence or divergence of

1. The Root Test

   The final test for convergence of a series is called the Root Test.
   
   
   

   The Root Test Let be a series with nonzero terms at the tail, then converges absolutely if              <  1 diverges if              >  1 If               =  1  then try another test.

   
   
   Example  
   
   Determine the convergence or divergence of

          
   
   We use the root test:   
   
          
   
   Hence the series diverges.

    

   Exercises  

   Determine the convergence or divergence of
   

   1. 
      

   2. 

Back to the Series Page

Back to the Math 107 Home Page

Back to the Math Department Home Page

e-mail Questions and Suggestions