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Find the value of $$\sum_{i=1}^{\infty}{\frac{\varphi(i)\cdot 7^i}{15^i-7^i}},$$where $\varphi(i)$ is the usual totient function.

Any ideas? I tried to coax a telescoping sum and even the closed form of $\varphi(i)$ but of course the totient function is not easily manipulated to create a telescoping sum. I put this into WA and got a result of about 1.6402.

Note: I am not very skilled at mathematics and a full solution (rather than hints) would be preferred.

  • Are you familiar with Abel summation? I think it is the most suitable way to solve this. – mertunsal Jun 28 '20 at 16:02
  • @mertunsal yes, ive heard of it. not familiar, per se, but ive seen it used before (in proving that $\sum_{n<x}{d(n)}=ln(x)+\gamma+O(\frac1x)$. –  Jun 29 '20 at 01:03

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