<s>[INST] Variation on Harmonic Series I'm trying to establish the convergence or divergence of the following variant of the harmonic series:
$$\frac{1}{1}+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}-\frac{1}{8}-\frac{1}{9}-\frac{1}{10}+\frac{1}{11}\cdots$$
Where the sign pattern has period 5, ie, it looks like this: ++---++---++---....
My thought has been to find a regrouping that diverges, since I only need to find one in order to show the series is divergent. I tried to bound this series below by increasing the denominator on the positive terms, and decreasing it on the negative terms to yield a series like $$\left(\frac{1}{2}+\frac{1}{2}\right)-\left(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\right)+\left(\frac{1}{7}+\frac{1}{7}\right)+ \cdots$$
but I'm pretty sure this will converge. I don't really know what approach to take next. A hint would be greatly appreciated! Thanks!
 [/INST] The sum of the first two is
$\dfrac1{5n+1}+\dfrac1{5n+2}
\lt \dfrac{2}{5n}
$
and the sum of the last 3 is
$\dfrac1{5n+3}+\dfrac1{5n+4}+\dfrac1{5n+5}
\gt \dfrac{3}{5n+5}
$.
Therefore
$\begin{array}\\
\dfrac1{5n+1}+\dfrac1{5n+2}
-(\dfrac1{5n+3}+\dfrac1{5n+4}+\dfrac1{5n+5})
&\lt  \dfrac{2}{5n}-\dfrac{3}{5n+5}\\
&=\dfrac{2 - n}{5 n (n + 1)}\\
&=\dfrac{2}{5 n (n + 1)}-\dfrac{1}{5  (n + 1)}\\
\end{array}
$
and the sum of this diverges
to $-\infty$.
</s>