# Thread: [SOLVED] Sum of Series...

1. ## [SOLVED] Sum of Series...

Find, as function of $x$, the sum of the infinite series and state its interval of convergence.

$x - 2x^2 + 4x^3 - 8x^4 + ... + (-1)^n 2^n x^{n+1} + ...$

Hey, I hope this is the right board..

How would I determine the answer? I think it's a combination of power series, (like $e^x$ + another series or something like that) but I'm just guessing.

Thanks.

2. Originally Posted by Solo
Find, as function of $x$, the sum of the infinite series and state its interval of convergence.

$x - 2x^2 + 4x^3 - 8x^4 + ... + (-1)^n 2^n x^{n+1} + ...$

Hey, I hope this is the right board..

How would I determine the answer? I think it's a combination of power series, (like $e^x$ + another series or something like that) but I'm just guessing.

Thanks.

$f(x)=\sum_{n=0}^{\infty}(-1)^{n}2^{n}x^{n+1}=x\sum_{n=0}^{\infty}(-2x)^n$

So this is a geometric series with $r=(-2x)$ so we get

$f(x)=x\cdot \frac{1}{1+2x}=\frac{x}{1+2x}$

Remember that geometric series converege when r< 1

3. Hello,

You're looking for :
$\sum_{n\geq 0} (-1)^n 2^n x^{n+1}$
Factor out x :
$=x\sum_{n\geq 0} (-1)^n 2^n x^n$
And you can see that this simplifies to :
$=x\sum_{n\geq 0} (-2x)^n$

Provided that $|-2x|<1 \Rightarrow |x|<\tfrac 12$, this is a convergent geometric series (multiplied by x)

Edit : muff...too late ><

Editē :
So this is a geometric series with r=(-2x) so we get

f(x)=x\cdot \frac{1}{1+2x}=\frac{x}{1+2x}

Remember that geometric series converege when r< 1
Well, actually you have to state the condition over x before writing what f(x) is

4. Ah yes!

You guys are legends. Thank you very much!

5. Originally Posted by Moo
Hello,

You're looking for :
$\sum_{n\geq 0} (-1)^n 2^n x^{n+1}$
Factor out x :
$=x\sum_{n\geq 0} (-1)^n 2^n x^n$
And you can see that this simplifies to :
$=x\sum_{n\geq 0} (-2x)^n$

Provided that $|-2x|<1 \Rightarrow |x|<\tfrac 12$, this is a convergent geometric series (multiplied by x)

Edit : muff...too late ><

Editē :

Well, actually you have to state the condition over x before writing what f(x) is
Thanks for keeping me honest Moo