∞                         ∞ 

   For what values of   x  does a Power Series Converge?    ∑ an (x – b)n   =  ∑ un  

                                                                                                 n=0                           n=0

 

 

In a Nut Shell:  Determination of the values of  x  for which a power series converges

is a three-step process.  Step 1 Use the ratio test (sometimes root test).  Step 2 is plot

the interval of convergence.  Step 3 is to check the end points of  x  on the real axis.

 

 

 

 

Here’s a reminder of the Ratio Test for the Power Series given at the top.

 

To test, take the limit:   lim   | (un+1)/ un |  =  lim   | (an+1)(x- b)n+1/ an (x – b)n|  

                                    n → ∞                            n → ∞

which becomes     lim   | {(an+1)/an }(x – b)|    =    P |x -b| 

                              n → ∞

 

  For convergence  P |x - b|   <  1     so  |x - b|  < 1/P  =  R  (radius of convergence)

 

Step 1    Find radius of convergence                                                                

                                           ∞                         

   Consider the example   ∑ (x – 1)n         Here   un  =   (x – 1)n     un + 1  =   (x – 1)n + 1 

                                          n=0                          

 

  lim   | (un+1)/ un |  =  lim   | x – 1 |   <   1           R  =  1  for convergence

  n → ∞                           n → ∞

                                                             or       - 1  <   (x – 1)  <   1

 

Step 2    Plot the interval of convergence on the real axis.

 

                                                       converges here

              __________________|_________|_________|______________   x

                                                 0                 1                 2

 

 

Step 3   Check end points of interval.   In this example the two end points are  x  =  2 

nd  x  =  0.  Put them in the original power series to obtain a series of constant terms. 

You then use any of the tests for an infinite series of constant terms to determine

convergence or divergence.

                       ∞                         

 For x  =  0,     ∑ ( – 1)n      diverges by nth term test or since series alternates

                      n=0                          

                       ∞                         

 For x  =  2,     ∑ ( 1 )n      diverges by nth term test

                      n=0                          

 

Result:  Interval of convergence  =  ( 0 , 2)        Radius of convergence  =  1.

 

Click here for other examples.

 



Copyright © 2017 Richard C. Coddington

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