Proof for a series given

Question

Prove that

$$\frac{z}{(1-z)^2} = \sum_{n=1}^{\infty} nz^n.$$

Do I need to do this by induction or by any other way?

Please help.

Answer 1

You do not want to try this with induction. You can't form an inductive hypothesis that means anything. You are only trying to prove one statement, not a whole family of statements. What you should try to do then is a direct proof.

Here's a hint: What is the series for $\dfrac{1}{1-z}$? How can you get your left hand side from this? (There are two easy ways to do this but one way is really, really ugly.)

Whenever you have a somewhat complicated series that "looks like" a basic series that you know, you should always see if you can manipulate the basic series to get the series you have. In this case, we can see that $\dfrac{1}{1-z}$ is somehow related to our left hand side so we should try to do something with this.

Answer 2

Hint: differentiate the geometric series.