Manipulating Power Series for Coefficient Extraction

[Amrator]

Homework Statement

By considering the power series (good for |x| < 1)

$\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + x^4 +...$

Describe how to manipulate this series in some way to obtain the result:

$\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2}$

Homework Equations

Maclaurin series?

The Attempt at a Solution

I was trying to somehow relate n to $\frac{x}{1-x}$ and then multiply the first series (with its index shifted) by that result. Problem is I couldn't find any relation. I basically don't even know where to start. I would appreciate a hint.

 

[blue_leaf77]

Have you tried taking the first derivative of the first series?

 

[Amrator]

Have you tried taking the first derivative of the first series?

No. I'll try that.

I guess my problem was understanding what they meant by "manipulating". Wouldn't taking the derivative of a series give me a whole new series? I don't see how that's "manipulating".

Thanks.

 

[blue_leaf77]

When you compare the compact forms of the function, you can't see any equivalence between them. They are indeed an entirely different functions. I think this problem just asks you to find a way to get the $x/(1-x)^2$ from $1/(1-x)$ by looking at their respective power series.

 

[Mark44]

No. I'll try that.

I guess my problem was understanding what they meant by "manipulating". Wouldn't taking the derivative of a series give me a whole new series? I don't see how that's "manipulating".

By "manipulating" the series, they mean applying some operation to it. Taking the derivative term-by-term definitely counts as manipulating the series.

 

[Amrator]

Alright, I took the first derivative and multiplied by x. Thanks guys.