<s>[INST] "Altered" Alternating Series Diverges or Converges? Consider the series
$$
1-\frac{1}{2} +\frac{1}{3} +\frac{1}{4} - \frac{1}{5} - \frac{1}{6} - \frac{1}{7} + \frac{1}{8} + \cdots + \frac{1}{11} + \cdots
$$
which alternates between a block of positives and a block of negatives, with the block sizes, 
$$
1, 1, 2, 3, 4, 5, 6, \dots
$$
and so on.
Does this series converge or diverge?
 [/INST] Let: $$
S=1-\frac{1}{2} +\frac{1}{3} +\frac{1}{4} - \frac{1}{5} - \frac{1}{6} - \frac{1}{7} + \frac{1}{8} + \cdots + \frac{1}{11} + \cdots
$$
For a given sequence of additions assume a constant denominator, specifically the smallest in the group. For a series of subtractions pick a common constant denominator, the largest in the group. This guarantees a maximum effect of additions and a minimum effect of subtractions, guaranteeing greater partial sums than the original series. 
Doing this we get:
$$S' = 1 -\frac{1}{2}+\frac{2}{3}-\frac{3}{7}+\frac{4}{8}-\frac{5}{17}+\frac{6}{18}...$$
Suppose the numerator  is $2k$ for some integer $k$. Then the denominator is $\frac{5k^2-5k+6}{2}$. This term is positive. The term directly to the left is negative with numerator $2k-1$ and denominator $\frac{5k^2-5k+4}{2}$.
So:
$$S'=1 + \sum_{k=1}^\infty \frac{4k}{5k^2-5k+6}-\frac{4k-2}{5k^2-5k+4}$$
This preserves the order of terms. 
If the two fractions are added, we get a second degree polynomial in the numerator and a 4th degree polynomial in the denominator for a terms on the order of $O(1/k^2)$. 
So $S'$ converges by comparison to $1/k^2$ and $S$ converges by comparison with $S'$. 
</s>