Funky Series 2

Calculus Level 4

Evaluate the infinite series

S=k=0(1)k (log20172018)kk! (log20172016)kS= \sum_{k=0}^{\infty} \frac{(-1)^k~(\log_{2017} 2018)^k}{k!~(\log_{2017} 2016)^k}

If ln(S)=loga(b)\ln(S) = -\log_a(b), where aa and bb are integers, find the smallest possible average of aa and bb.

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