A convergent regrouping of a series which tends to 0 implies convergence

Question

A while ago I was trying to solve the problem of whether the following series converges: $ 1 -1/2 -1/3 +1/4+1/5-1/6-1/7+1/8+1/9...$ My solution was to repeat a similar argument to the common proof od the Alternating series test where I show that the sequence of partial sums of positive terms converges and the sequence of partial sums of the negative terms converges, then using that show that the partial sums converge. However, the above series can easily be grouped into an alternating series. My question is, is the convergence of a single regrouping, combined with the fact that the terms of the original series individually go to 0 enough to imply convergence?

Answer 1

It is as long as there is some $k$ such that each group is of size at most $k$. Lets say the $k$-th group starts at position $a_k$.

Then partial sum of the original series is $\sum\limits_{i=1}^n x_i = \sum\limits_{k = 1}^m \sum\limits_{i = a_k}^{a_{k + 1} - 1} x_i + \sum\limits_{i=a_{m + 1}}^n x_i$, where $m$ is such that $a_{m + 1} < n \leqslant a_{m + 2}$. The first sum is a partial sum of regrouped series, and the second goes to zero, as it has at most $k$ addends and $x_i \to 0$.