Question

How do you find a power series representation for `(10x)/(14+x)` and what is the radius of convergence?

Answer 1

`10( x/14 -(x/14)^2 +(x/14)^3-......oo)`

Radius of convergence 14

Explanation:

`(10x)/(14+x)= 10(x/(14+x))`

=`10(1- 14/(14+x))`

=`10 - 10(1/(1+x/14))`

Now comparing with the geometric power series `1/(1-x)= 1 +x +x^2 ....oo` and writing `-x/14` for x, the series would be,

`10-10(1-(x/14) +(x/14)^2 -(x/14)^3 +.....oo)`

`=10( x/14 -(x/14)^2 +(x/14)^3-......oo)`

The implied condition of a convergent geometric series represented by `1/(1-x)` is -1< x <1, hence in the present case it would be -1< -x/14 <1 0r -14 < -x <14 Or 14> x >-14. Thus the radius of convergence is 14