1. ## find the sum

find the sum of the given series:

heres the equation: http://img19.imageshack.us/img19/559/untitledyc9.jpg

I was told it could be done by first finding a and b using that 2n +1 = a(n+1) + b

and given as a hint that E(n + 1)x^n is the derivateive of a known power series already.

thankyou.

2. Find the interval of absolute convergence and then.
$\displaystyle \sum_{n=0}^{\infty}(2n+1)x^n$
Expand,
$\displaystyle 1+3x+5x^2+7x^3+...$
$\displaystyle (2x+4x^2+6x^3+...)+(1+x+x^2+x^3+...)$
$\displaystyle 2x(1+2x+3x^2+...)+(1+x+x^2+...)$. (*)
Okay, now we notice a very important thing.

First we know that,
$\displaystyle 1+x+x^2+x^3+...=\frac{1}{1-x}$
Then, take derivative,
$\displaystyle 1+2x^2+3x^2+...=\frac{x}{(1-x)^2}$.
Substitute that into (*),
$\displaystyle 2x\cdot \frac{x}{(1-x)^2}+\frac{1}{1-x}$
$\displaystyle \frac{2x^2}{(1-x)^2}+\frac{1-x}{(1-x)^2}$
$\displaystyle \frac{2x^2-x+1}{(1-x)^2}$

3. thanks for the help,

but where does (2x + 4x^2 + 6x^3 +...) + (1 +...

come from?

also, what is the sum?

thankyou.

4. Originally Posted by rcmango
thanks for the help,

but where does (2x + 4x^2 + 6x^3 +...) + (1 +...

come from?

also, what is the sum?

thankyou.
Well, add them you get the same thing. No?