Root Test for Series

 

 

 

 

Theorem ( Root Test for Series )

                  .  

(1)     if  r < 1,  then  { S_n } converges.

(2)     If r > 1,  then { S_n } diverges.

(3)     When r=1,  the test gives no information.

 

    Proof:

                         . 

                    So,   for any  > 0,  there exist a corresponding K such that

 

                                        <   if  n> K .

                            (r-  ) <   < (r+  )     for  n > K

                                  (r-  )ⁿ  < | c_n | < (r+  )ⁿ      for n > K

 

(1)     if  r < 1,  we can choose  such that  (r+  ) < 1 .  Then

 

           =

                       <  <  when m goes to .

 

      <  .  Thus, {S_n} converges.

 

(2)     if r > 1,  we can choose  such that (r-  ) > 1 .  Then

 

               1<  ( r-  )ⁿ <  | c_n |   for n > K

 

     There is no hope that   .

    The sequence diverges.

 

 

 

 

 

 

Example:    .      < 1 .   So,  { S_n } converges.

 

 

 

 

       This theorem will be useful once we prove   .  Before that, we only browse some easy examples.

 

 

 

Example: 

 

                     The series converges because  .