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Power Series

Posted on 2nd September 2010 by admin in bicycle - Tags: diy, electronics, power series convergence, power series examples, power series expansion, power series representation, power series square root, series, tv, video

Power Series

Find the radius of convergence of the power series?

How would I go about solving this problem:

Suppose that (10x)/(14+x) = the sum of CnX^(n) as n=0 goes to infinity

C1=
C2=

Find the radius of convergence R of the power series.
R=

--What does it mean by C1, C2, C3,... ?
I don't understand what its asking for. Can some one please walk me through this so that can I understand "Representing Functions as Power Series".

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.

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Power Series Representation of Functions