Does $ 1 + 1/2 - 1/3 + 1/4 +1/5 - 1/6 + 1/7 + 1/8 - 1/9 + ...$ converge?

Question

Does $ 1 + 1/2 - 1/3 + 1/4 +1/5 - 1/6 + 1/7 + 1/8 - 1/9 + ...$ converge?

I know that $(a_n)= 1/n$ diverges, and $(a_n)= (-1)^n (1/n)$, converges, but given this pattern of a negative number every third element, I am unsure how to determine if this converges.

I tried to use the comparison test, but could not find sequences to compare it to, and the alternating series test doesn't seem to work, because every other is not negative.

Answer 1

Hint: $1 + 1/2 - 1/3 > 1$, $1/4 + 1/5 - 1/6 > 1/4$, $1/7 + 1/8 - 1/9 > 1/7$, ...

Answer 2

Let $s_n$ be the $n$th partial sum of the series. If the series converges, then the sequence $\{s_n\}$ is bounded.

However, observe that $s_4>1+\frac{1}{4}$, $s_7> 1+\frac{1}{4}+\frac{1}{7}$, and in general $$ s_{3m+1}>\sum_{k=0}^m\frac{1}{3k+1} $$ Since $\sum_{k=0}^{\infty}\frac{1}{3k+1}$ diverges, this shows that the sequence $\{s_n\}$ is not bounded, so the series diverges.