$\displaystyle \sum^{\infty}_{n = 0}(2n+1).4^{n}$
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!
$\displaystyle \sum^{\infty}_{n = 0}(2n+1).4^{n}$
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
$\displaystyle \displaystyle \sum^{\infty}_{n = 0}(2n+1)x^{n}\bigg|_{x=.4} = 2\sum_{n=0}^{\infty}nx^n+\sum_{n=0}^{\infty}x^n=2x \sum_{n=0}^{\infty}\frac{d}{dx}x^n+\sum_{n=0}^{\in fty}x^n$
$\displaystyle \displaystyle 2x\frac{d}{dx}\left( \frac{1}{1-x}\right)+\frac{1}{1-x}=\frac{2x}{(1-x)^2}+\frac{1}{1-x}=\frac{1+x}{(1-x)^2}$
Now just plug in $\displaystyle x=0.4$
This gives
$\displaystyle \displaystyle \frac{1.4}{(.6)^2}=3.\bar{8}$