Let $S_0=1$, and let $S_{n+1}$ be the sum of $1/k+1/(k+1)+1/(k+2)\cdots 1/(k+n)$, for the integer $k$ such that $S_{n+1}$ is maximal while $S_{n+1}
Does $\sum_{k=1}^{\infty}S_k(-1)^{k}$ converge?
Minor addition:
What is $\lim_{n\to\infty} S_n ?$
And is it possible to obtain a formula for $k$ as a function of $n$?
And the series $1-(1/2+1/3)+(1/4+1/5+1/6)-(1/7+1/8+1/9+1/10)+\cdots$ converges by the alternating series test, does it have a closed form for its sum?