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What is the sum of the following series?

$$\sum_{N=1}^\infty\frac{N}{100} \cdot\left(\frac{99}{100}\right)^{N-1}$$?

This looks like a geometric series but I think the infinite sum formula wouldn't work here as there's an $\frac{N}{100}$ in front of the series?

I think the series converges to $100$, but I'm not sure how to get there

Clement C.
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syndee
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    Welcome to this site! Here is a comment to get you started. It's actually easier to forget about the $99/100$ and look at the function $f$ defined by $$f(x) = \sum_{n=1}^\infty n x^{n-1}$$ (what you want is the number $\frac{1}{100}f(99/100)$; but functions are often nicer to manipulate than numbers, since then you can use calculus, etc.). Now, consider $g$ defined by $$g(x)=\sum_{n=1}^\infty x^n$$ (1) Can you compute a nicer expression for $g(x)$? (2) Can you find a way to relate $f$ and $g$ (think: differentiation)? (3) Can you use that to get a nicer expression for $f(x)$? – Clement C. Feb 16 '20 at 00:41
  • Equivalent questions have been asked and answered here many times. – Gerry Myerson Feb 16 '20 at 01:54

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