Convergence of Complex Series: Finding the Sum and Radii of Convergence

[Redsummers]

Homework Statement

$$\sum_{n=0}^\infty} (n+1)z^n$$

We have to find for which values of z is the series converging and also, we should find the sum of such a complex series.

Homework Equations

$$R = \frac{1}{\lim a_n^(1/n)}$$

$$r<\rho<R$$

$$q = \frac{r}{\rho}$$

The Attempt at a Solution

First of all, I searched for a series expansion similar to our sum (I first thought it would be log (n+1) ) but it wasn't the case.

Then, I looked up for my lecture notes on uniform convergence and continuity, and I found these equations stated above. I think this should definitely help since it can also be applied to complex series.

The problem, though, is that I don't really know how to use such radii of convergence. I also wrote down the factor q, since it rather seems that to solve this sum I will most likely need the geometric series (sum q^n = 1/(1-q) ). So my problem really is how to deal with the (n+1) factor, otherwise, if it was just z^n it would just converge to 1/(1-z) for all |z| < 1 = r.

I would appreciate any help.

 

[LCKurtz]

Hint: Does that look like the derivative of anything?

 

[Redsummers]

Yeah, I guess that just looks as if we took the derivative of z^(n+1), but I don't see how is this going to help us anyway *shrug*.

Maybe I'm not counting some fundamental definition of integrals, series and its relations?

 

[Redsummers]

Oh got it!
You were right about looking at the derivative. It was just that the sum of z^n (from 0 to inf) equals to a geometric series, then, if we take the derivative of this geometric series, we get exactly our sum. I.e. taking the derivative of the geometric series sum (1/(1-z)) will be our result, which ended up to be 1/(z-1)^2

Thanks!

 

[LCKurtz]

Oh got it!
You were right about looking at the derivative. It was just that the sum of z^n (from 0 to inf) equals to a geometric series, then, if we take the derivative of this geometric series, we get exactly our sum. I.e. taking the derivative of the geometric series sum (1/(1-z)) will be our result, which ended up to be 1/(z-1)^2

Thanks!

Be careful about the first term and your sign when you differentiate.

 

[Redsummers]

Oh, sure. In that case though, (z-1)^2 = (1-z)^2. But for odd powers we should put a minus sign. So actually, if you think of it, we can come up with a nice formula, such as:

$$\sum_{n=0}^\infty} (n+1)(n+2)...(n+\alpha)z^n = \frac {\alpha !}{(1-z)^{\alpha+1}}$$