MHB What is the sum to infinity of this nearly geometric series?

Poirot1
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Find the sum to infinity of the series $1 +2z +3z^2+4z^3+...$
 
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Integrate it, find the sum of the integrated series, then differentiate it.
 

Thread 'Apparent counter-example to Cauchy-Goursat theorem (Complex Analysis)'

I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...
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