General Term of specific Alternating Series

Question

Recently I think about series below and I wonder if there is away to write the general term of it... $$ 1-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4} + \dots $$ The problem to me is changing the sign. since number of changing sign each time is change and increased by one time with the pervious time ... if they was equal I could find it using floor function... Any way... Any help to find the general time will be appreciated .

Answer 1

$\sum_{i=1}^\infty \left((-1)^{i+1}i\sum_{k=1}^i\left(\frac{1}{i}\right)\right)$

Answer 2

Ignoring the sign, $\dfrac1n$ will first appear after $$1+2+\cdots+n-2+n-1=\frac{n(n-1)}2$$

$\dfrac1{n+1}$ will first appear after $\displaystyle\frac{n(n+1)}2$

So, if $\displaystyle\frac{n(n-1)}2<m\le\frac{n(n+1)}2,$ $$t_m=(-1)^{m-1}\frac1n$$