Calculate a meromorphic function on $\mathbb{C}$ s.t. $g(z) = \sum_{n\geq0}(n+1)T^n$

Question

I have the following power series $f(T)=\sum_{n\geq0}(n+1)T^n$ and I have to calculate a meromorphic function $g$ on $\mathbb{C}$ such that $\forall z \in B_1(0), g(z)=f(z)$

I have tryed the following:

$f(z) = \sum_{n\geq0}nz^n + \sum_{n\geq0}z^n$

and I used that

$z\sum_{n\geq0}nz^{n-1} = z \frac{1}{(1-z)^2}$ and $\sum_{n\geq0}z^{n-1} = \frac{1}{1-z}$ because $|z|\leq1$

Then $f(z)= \frac{z}{(1-z)^2} + \frac{1}{1-z} = \frac{1}{(1-z)^2}$

That I know is wrong but I could not find the error.

Can anyone help me? Thank you in advanced.

Answer 1

As mentioned in a comment, you computation is correct, but you can make it simpler, using the formal derivative in $\mathbf C[[T]]$:

$$\sum_{n\ge 0}(n+1)T^n=\sum_{n\ge 1}nT^{n-1}=\Bigl(\sum_{n\ge 0}T^n\Bigr)'=\Bigl(\frac1{1-T}\Bigr)'=\frac1{(1-T)^2}.$$