Let $\sum a_n$ be a series of reals which is constructed as follows -
The block $\frac{1}{k},-\frac{1}{2k},-\frac{1}{2k}$ repeats $2k+1$ times. That is, $\sum a_n = 1 - \frac{1}{2} - \frac{1}{2} $ repeating 2 times $+ \frac{1}{2} - \frac{1}{4} - \frac{1}{4}$ repeating 5 times and so on.
My question is, is the series convergent? The individual blocks sum up to 0, but are we allowed to group terms before knowing the convergence?
I tried Dirichlet's Test, but the partial sum of $1,-1,-1,1,-1,-1,...$ is not bounded since there are more negative terms than positive ones.
How can we determine the convergence?
First of all, let's look at the sequence of partial sums: $$ 1, \frac12, 0, 1, \frac12, 0,\frac12, \frac14, 0,\frac12, \frac14,0,\ldots,\frac1k, \frac1{2k},0,\ldots $$ Even without grouping terms, that looks to me like it converges to $0$ (regardless of how many times each block of $3$ repeats before moving on to the next, as long as it's a finite number of times). And once you go through the formal $\epsilon$-$N$ argument, you will see that it indeed converges to $0$.