Does the series $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}+\cdots $ converge or diverge? Does the series $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}+\cdots $ converge or diverge?

\begin{align}
& \underbrace{1+\frac{1}{2}-\frac{1}{3}}_{\ge 1} + \underbrace{\frac{1}{4}+\frac{1}{5}-\frac{1}{6}}_{\ge 1/4} + \underbrace{\frac{1}{7}+\frac{1}{8}-\frac{1}{9}}_{\ge 1/7} + \underbrace{\frac 1 {10} + \frac 1 {11} - \frac 1 {12}}_{\ge 1/10} + \cdots \\[10pt]
\ge {} & 1 + \frac 1 4 + \frac 1 7 + \frac 1 {10} + \frac 1 {13} + \cdots \\[10pt]
\ge {} & \frac 1 3 + \frac 1 6 + \frac 19 + \frac 1 {12} + \frac 1 {15} + \cdots \\[10pt]
= {} & \frac 1 3 \left( 1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5 + \cdots \right) = \infty.
\end{align}
