Sum of Gaps 2

Calculus Level 3

Given that π=4k=1(1)k+12k1,\displaystyle \pi = 4\sum_{k=1}^{\infty} \frac{\left(-1\right)^{k+1}}{2k-1}, what is the value of

n=1(π4k=1n(1)k+12k1)?\sum_{n=1}^{\infty} \left( \pi - 4\sum_{k=1}^{n} \frac{\left(-1\right)^{k+1}}{2k-1} \right)?


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