Oldest problem on Brilliant

Number Theory Level 5

\[A=\displaystyle \sum _{ n=0 }^{ \infty }{ \frac {(-1)^n \tau(2n+1) }{2n+1 } }\]

Given that \(\tau(N)\) denotes the number of positive integer divisors of \(N\). Find \(\left\lfloor 100000A \right\rfloor\).


I'm quite surprised by how it converges (At least I think it does, I'm a noob at determining if a series converge), considering \(\displaystyle \sum _{ n=1 }^{ \infty }{ \frac {(-1)^n\tau(n) }{n } }\) diverges, as well as the simplicity in the solution.

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