Inspired by Rohit Ner

Calculus Level 5

limNn=1N((1)n+1(2k=02n(1)k(2nk)))\displaystyle\lim_{N \rightarrow \infty} \sum_{n=1}^{N} \left( (-1)^{n+1} \left( 2 - \sum_{k=0}^{2n} \dfrac{(-1)^{k}}{\dbinom{2n}{k}} \right) \right)

If the above expression equals aln(b),a - \ln(b), where aa and bb are positive integers, then find a3+b3.a^{3} + b^{3}.

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