Consider this special form of an alternating sum
$$s=1-(\frac{1}{2})+(\frac{1}{3}+\frac{1}{4})-(\frac{1}{5}+\frac{1}{6}+\frac{1}{7})+(\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{11})-(\frac{1}{12}+...+\frac{1}{16})+...$$
Notice that the ranges of equal signs increase linearly.
I wonder if the sum is convergent or not, and if it is convergent what is the result.
Edit by Rushba Mehta:
The explicit form of the series is $$1 - \sum\limits_{n=2}^\infty (-1)^{\lfloor\frac{\sqrt{8n-15}-1}2\rfloor}\cdot\frac1n$$
My Edit
I find that the sum can be written as
$$s=1+\sum_{n=0}^\infty (-1)^{n+1}(H_{n(n+3)/2+2}-H_{(n^2+n+2)/2})$$
Where $H_{k}=1+1/2+1/3+...+1/k$ is the harmonic number.
Convergence follows from the asymptoptic expansion of $H_{k}$ which leads to a series similar to that of log(2).