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 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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