# Thread: Can you sum this series analytically?

1. ## Can you sum this series analytically?

$\sum^{\infty}_{n = 0}(2n+1).4^{n}$

It's in reference to this (un)real world problem: A rat is put in a box with two exit doors.? - Yahoo! Answers

Thanks for any help!

2. Originally Posted by NowIsForever
$\sum^{\infty}_{n = 0}(2n+1).4^{n}$

It's in reference to this (un)real world problem: A rat is put in a box with two exit doors.? - Yahoo! Answers

Thanks for any help!
Consider the series

$\displaystyle \sum^{\infty}_{n = 0}(2n+1)x^{n}\bigg|_{x=.4} = 2\sum_{n=0}^{\infty}nx^n+\sum_{n=0}^{\infty}x^n=2x \sum_{n=0}^{\infty}\frac{d}{dx}x^n+\sum_{n=0}^{\in fty}x^n$

$\displaystyle 2x\frac{d}{dx}\left( \frac{1}{1-x}\right)+\frac{1}{1-x}=\frac{2x}{(1-x)^2}+\frac{1}{1-x}=\frac{1+x}{(1-x)^2}$

Now just plug in $x=0.4$

This gives

$\displaystyle \frac{1.4}{(.6)^2}=3.\bar{8}$

3. I wrote the wrong problem. It should have been

$\displaystyle \sum^{\infty}_{n = 0}(2n+3)x^{n}=\frac{3-x}{(1-x)^2}$

Which is 65/9 when x = .4, and is in accord with the WolframAlpha result.

Thanks!