Math Help - Power Series, Function Representation

1. Power Series, Function Representation

Hi there,
I am trying to Verify that $f(x)=\frac{1}{3+2x}$
has power represenation of the sum from n=0 to infinity of
$\sum_{n=0}^{\infty}\frac{(-1)^n2^n}{3^{n+1}}*x^n$
I am aware of the differentiation and integration rules of the series, will I need those to complete the problem?
Any direction would be greatly helpful.
Thank you!!

2. Originally Posted by matt.qmar
Hi there,
I am trying to Verify that $f(x)=\frac{1}{3+2x}$
has power represenation of the sum from n=0 to infinity of
$\sum_{n=0}^{\infty}\frac{(-1)^n2^n}{3^{n+1}}*x^n$
I am aware of the differentiation and integration rules of the series, will I need those to complete the problem?
Any direction would be greatly helpful.
Thank you!!
Your series outputs $f^n(0)$ correctly, but it seems you are missing the division by n! aren't you?

3. Originally Posted by matt.qmar
Hi there,
I am trying to Verify that $f(x)=\frac{1}{3+2x}$
has power represenation of the sum from n=0 to infinity of
$\sum_{n=0}^{\infty}\frac{(-1)^n2^n}{3^{n+1}}*x^n$
I am aware of the differentiation and integration rules of the series, will I need those to complete the problem?
Any direction would be greatly helpful.
Thank you!!
$\frac{1}{3+2x} = \frac{\frac{1}{3}}{1 - \left(-\frac{2x}{3}\right)} = \frac{1}{3} - \frac{2x}{3^2} + \frac{2^2x^2}{3^3} - \frac{2^3x^3}{3^4} + ...$

4. Skeeter's point is that the geometric series $\sum_{n=0}^\infty ar^n= \frac{a}{1- r}$. Since $\frac{1}{3+ 2x}= \frac{\frac{1}{3}}{1- (-\frac{2}{3})}$, this is the sum of a geometric series with $a= \frac{1}{3}$, $r= -\frac{2}{3}$.