Power Series, Function Representation

**matt.qmar** · November 9th 2009, 04:50 PM

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!!

**Jameson** · November 9th 2009, 06:12 PM

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?

**skeeter** · November 9th 2009, 06:16 PM

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!!

$\frac{1}{3+2x} = \frac{\frac{1}{3}}{1 - \left(-\frac{2x}{3}\right)} = \frac{1}{3} - \frac{2x}{3^2} + \frac{2^2x^2}{3^3} - \frac{2^3x^3}{3^4} + ...$

**HallsofIvy** · November 10th 2009, 04:26 AM

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 $a= \frac{1}{3}$ , $r= -\frac{2}{3}$ .

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