Any method

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients.

b. If possible, determine the radius of convergence of the series.

Solution 59E

Step 1:

a) In this problem we need to find the first four nonzero terms of the taylor series centered at ‘0’ for the function f(x) = by using the taylor series :

We know that , , for -1 < x < 1

Consider , = , since

=

By , using for -1 < x< 1 ,we get

= 1 -2()+3, for -1 < < 1

= 1 -2+3,for -1 < < 1

Therefore , the first four nonzero terms of the taylor series centered at ‘0’ for the function are 1 -2+3 .

That is , 1, - 2 , 3 ,.

First term = 1

Second term = - 2

Third term = 3

Fourth term =

Step 2:

Power series using the summation notation of the function f(x) = = , centered at a = 0 is;

f(x) = = 1 -2+3

Therefore ,

Power series using the summation notation is ; ,for -1 < < 1