## Find Sum of Series Using Power Series

1. The problem statement, all variables and given/known data

Find the sum of sigma (n=1 to infinity) (-1)^(n-1) * [n/(4^(n-1))] using the power series 1/(1+x) = sigma (n=0 to infinity) (-1)^n * x^n.

2. Relevant equations
1/(1+x) = 1 - x + x^2 - x^3 + x^4 + ... + (-1)^n * x^n

3. The attempt at a solution

The problem suggested to differentiate both sides and then substitute 1/4 for x.

I am totally confused and don't even know where to begin.
 PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks
 Recognitions: Homework Help You know how to differentiate right: $$\frac{d}{dx}\frac{1}{1+x}=-\frac{1}{(1+x)^{2}}$$ I am sure you know how to differentiate $x^{n}$, what do you get if you do this?

 Quote by hunt_mat You know how to differentiate right: $$\frac{d}{dx}\frac{1}{1+x}=-\frac{1}{(1+x)^{2}}$$ I am sure you know how to differentiate $x^{n}$, what do you get if you do this?
d/dx (x^n) = n * x^(n-1)

Recognitions:
Homework Help

## Find Sum of Series Using Power Series

Right, so if you differentiate the series term by term (which you're allowed to do as long as you're within the radius of convergence), what do you get?

 Quote by hunt_mat Right, so if you differentiate the series term by term (which you're allowed to do as long as you're within the radius of convergence), what do you get?
I'm not sure I understand the question.

Which series?

Maybe I'm just not understanding the original question.

1/(1+x) can be rewritten as sum (n=0 to infinity) of (-1)^n * x^n.

I believe the question is asking me to take the sum (n=1 to infinity) of (-1)^(n-1) * n/4^(n-1) and come up with a small simple equation to determine its sum using the pattern of 1/1+x as a template. Yes?

I know that if I take x^n and differentiate it I get n/x^n-1, which looks exactly like my n/4^(n-1). Does that mean that the sum of my series is nothing more than 1 - 1/4 + 1/16 - 1/64? Do I need to make an adjustment to my n values because my sum starts @1 and the original starts @0?
 Recognitions: Homework Help Differentiating the series: $$\sum_{n=0}^{\infty}(-1)^{n}x^{n}$$ is $$\sum_{n=1}^{\infty}(-1)^{n}nx^{n-1}$$ Now what is the above equal to? (hint, we have derived it before)

Recognitions:
Gold Member
Homework Help
Staff Emeritus
 Quote by Rapier 2. Relevant equations 1/(1+x) = 1 - x + x^2 - x^3 + x^4 + ... + (-1)^n * x^n 3. The attempt at a solution The problem suggested to differentiate both sides and then substitute 1/4 for x. I am totally confused and don't even know where to begin.
Do exactly what was suggested. What do you get if you differentiate the lefthand side? What do you get if you differentiate the righthand side? Then plug in x=1/4. What do you get? How is it related to the original question?

 Quote by hunt_mat Differentiating the series: $$\sum_{n=0}^{\infty}(-1)^{n}x^{n}$$ is $$\sum_{n=1}^{\infty}(-1)^{n}nx^{n-1}$$ Now what is the above equal to? (hint, we have derived it before)
Since I differentiated the right side, I have to differentiate the left side, neh?

So that will equal -1/(1+x)^2?

I'm trying not to feel really stupid, and I'm just so frustrated. I greatly appreciate all your help.

EDIT: When you differentiate do you change the starting value for n? Since I'm not differentiating n (-1)^n stays (-1)^n, shouldn't it?

EDIT EDIT: Sum = - 16/25?
 Recognitions: Homework Help Correct, now following your suggestion, set x=1/4, what will the series look like?

 Quote by hunt_mat Correct, now following your suggestion, set x=1/4, what will the series look like?
This is my differentiation:
$$\sum_{n=1}^{\infty}(-1)^{n}nx^{n-1}$$

The series was listed as:
$$\sum_{n=1}^{\infty}(-1)^{n-1}nx^{n-1}$$

Does the exponent on the (-1) change when differentiated?

1 -1/2 + 3/16 - 1/16 + ...
 Recognitions: Homework Help The idea here us to note that $(-1)^{n-1}=(-1)^{n+1}$ and you have a minus coming from the LHS.

 Quote by hunt_mat The idea here us to note that $(-1)^{n-1}=(-1)^{n+1}$ and you have a minus coming from the LHS.
OH! I see that! If I divide (-1)^n by -1, I get (-1)^n-1. DUH!

That is how it comes up with being 1/(1+x)^2.

That makes sense.
 Recognitions: Homework Help So, now you can tell us what the sum of the series is?

 Quote by hunt_mat so, now you can tell us what the sum of the series is?
16/25!!
 Recognitions: Homework Help Well done!!!

 Quote by hunt_mat Well done!!!
Thank you SO much for your patience and help. Not only did I understand what I did I can do it on other problems!