# Thread: Power series representation????

1. ## Power series representation????

Can anyone please help me???

It asks to find the power series representation and radius of convergence for

f(x) = ln(a - x) for a>0
As an answer I got the series 1 to infinity of
__x^n___
a^n+1
I don't know if it is right and the radius of convergence I got was a but I don't know if that is right

f(x) = __x^3___
(1-4x)^2
I have no idea how to do this one...

Please help me

2. Originally Posted by Blackberry410
Can anyone please help me???

It asks to find the power series representation and radius of convergence for

f(x) = ln(a - x) for a>0
As an answer I got the series 1 to infinity of
__x^n___
a^n+1
I don't know if it is right and the radius of convergence I got was a but I don't know if that is right

f(x) = __x^3___
(1-4x)^2
I have no idea how to do this one...

Please help me
Let

$f(x)=\ln(a-x)$ then
n
$f'(x)=\frac{-1}{a-x}=-\frac{1}{a} \cdot \frac{1}{1-\frac{x}{a}}$

using the defintion of the geometric series we get

$f'(x)=-\frac{1}{a} \sum_{n=0}^{\infty} \left( \frac{x}{a}\right)^n= -\sum_{n=0}^{\infty} \frac{x^n}{a^{n+1}}$

integrating both sides we get

$f(x)= \ln(a-x)= \left( - \sum_{n=0}^{\infty} \frac{x^{n+1}}{a^{n+1}(n+1)} \right) +C$

To find the constant C we evaluate

$f(0)= \ln(a)= \left( - \sum_{n=0}^{\infty} \frac{0^{n+1}}{a^{n+1}(n+1)} \right) +C \iff \ln(a) = C$

$f(x)= \ln(a-x)= \ln(a) - \sum_{n=0}^{\infty} \frac{x^{n+1}}{a^{n+1}(n+1)}$

Since this based on the geometric series it will converge when r < 1 so

$|\frac{x}{a}| < 1 \iff -1 < \frac{x}{a} < 1 \iff -a < x < a$

3. ## hint for the 2nd part

let $g(x)=\frac{1}{4} \cdot \frac{1}{1-4x}=\sum_{n=0}^{\infty}4^{n-1} x^n$

note that:

$\frac{x^3}{(1-4x)^2} = x^3 \cdot g'(x)$