I do not know where to start, any hints are welcome.
Answer
Note that ∞∑n=0zn=11−z
for |z|<1.
Differentiating (1) and multiplying by z (this is legitimate since for any r<1, ∑∞n=1nzn−1 converges uniformly for |z|≤r<1) yields
zddz∞∑n=0zn=∞∑n=0nzn=∞∑n=1nzn=z(1−z)2
for |z|<1.
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