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

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The series $1 + 2z + 3z^2 + 4z^3 + ...$ can be analyzed by integrating and differentiating to find its sum to infinity. The series can be expressed in terms of a function, which is then integrated to yield a new series. The sum of the integrated series is derived, and differentiation is applied to return to the original series form. The final result reveals the sum to infinity in terms of $z$. This method effectively demonstrates the relationship between integration, differentiation, and series summation.
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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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