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'Generally Geometric' printed from https://nrich.maths.org/
The sum of the infinite geometric series $1 + x + x^2 + x^3 + \cdots$ and the binomial series are well known.
How are the two related?
Show that $$\sum_{n=0}^\infty n x^n = {x\over(1-x)^2}$$ and find $$\sum_{n=0}^\infty n^2x^n.$$ Outline a method for finding $$\sum_{n=0}^\infty n^kx^n$$ where you do not have to carry out this computation beyond $k=2$.
Experiment with other expansions to try to find out the values for other interesting series.