Summation 2017

Algebra Level 3

∑ n = 1 ∞ n 201 7 n = a b \sum_{n=1}^\infty \frac{n}{2017^n}=\frac{a}{b} Find the sum of coprime positive integers a a and b b .


The answer is 4066273.

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Adhm Mohamed by Adhm Mohamed , 19, Egypt

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1 solution

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Apr 28, 2018

Let 1 2017 + 2 201 7 2 + 3 201 7 3 + . . . = N \frac{1}{2017}+\frac{2}{2017^2}+\frac{3}{2017^3}+...=N ,then 2017 N = 1 + 2 2017 + 3 201 7 2 . . . 2017N=1+\frac{2}{2017}+\frac{3}{2017^2}... , subtract and get 2016 N = 1 + 1 2017 + 1 201 7 2 + . . . = 2017 2016 , N = 2017 201 6 2 2016N=1+\frac{1}{2017}+\frac{1}{2017^2}+...=\frac{2017}{2016},N=\frac{2017}{2016^2}

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