(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[tex]\sum_{n=0}^\infty} (n+1)z^n[/tex]

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.

2. Relevant equations

[tex]R = \frac{1}{\lim a_n^(1/n)}[/tex]

[tex] r<\rho<R [/tex]

[tex] q = \frac{r}{\rho} [/tex]

3. 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 onuniform 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 factorq, 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 justz^nit would just converge to 1/(1-z) for all |z| < 1 = r.

I would appreciate any help.

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# Convergence of complex series

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