Power Series

Find the radius of convergence of the power series?

How would I go about solving this problem:

An infinite geometric power series...
sum(a*r^n) n: 0 --> infinity = a / (1 - r) , which converges if |r| < 1

So we want to try and write the given expression in this form so we can determine the what the terms of the sum are.

(10x)/(14+x) = 10 + ( (-140) / (x+14) )=
10 + ( (-10) / (1 - (-x/14) ) )

So comparing term by term with the sum of the infinite geometric series...
a = -10
r = -x/14

(10x) / (14+x) = 10 + sum(-10*(-x/14)^n ) n:0-->infinity

This is not exactly in the form asked for as we have a constant outside of the geometric sum, so take it or leave it.

C1, C2 , etc.. simply denote the coefficients of the 1st,2nd power of x respectively.
The n=1 term of the series is -10*(-x/14) so
C1 = 10/14 = 5/7
The n=2 term of the series is -10*(x^2/196) so
C2 = -10/196 = -5/98

For the radius of convergence we check the common ratio against the condition |r| < 1
Our common ratio is (-x/14)
| -x / 14 | < 1
implies
| x | < 14
Our radius of convergence about zero is 14.

Power Series Representation of Functions

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