Sum to infinity?

Algebra Level 4

$\large \frac{1}{2017} + \frac{2}{2017^2} + \frac{3}{2017^3} +\dots = \frac{a}{b^2}$

If $a$ and $b$ are positive integers satisfy the equation above , where $a$ is square-free , find $a+b$ .

The answer is 4033.

1 solution

Chew-Seong Cheong
Sep 21, 2017

$\begin{aligned} S &=\sum_{\color{#3D99F6}k=1}^\infty \frac k {2017^k} =\sum_{\color{#D61F06}k=0}^\infty \frac k {2017^k} \\ & =\sum_{\color{#D61F06}k=0}^\infty \frac {k+1}{2017^{k+1}} \\ & = \frac 1{2017} \left(\sum_{\color{#D61F06}k=0}^\infty \frac k {2017^k} + \sum_{\color{#D61F06}k=0}^\infty \frac 1{2017^k} \right) \\ & = \frac 1{2017} \left(S + \frac {2017}{2016} \right) \\ \implies S &= \frac {2017}{2016^2} \end{aligned}$

$\implies a+b=2017+2016=\boxed{4033}$

Sum to infinity?

The answer is 4033.

1 solution

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