Check if the series $\sum_{n=1}^\infty\frac{(-1)^\frac{n(n-1)}{2}}{\sqrt n}$ converges

Question

Check if the series $\sum_{n=1}^\infty\frac{(-1)^\frac{n(n-1)}{2}}{\sqrt n}$ converges

So I think that the series is conditionally convergence that what I did so far

$$\sum_{n=1}^\infty\vert\frac{(-1)^\frac{n(n-1)}{2}}{\sqrt n}\vert>\frac{1}{n}$$ therefore by the comparison test the series diverges but to prove the series $\sum_{n=1}^\infty\frac{(-1)^\frac{n(n-1)}{2}}{\sqrt n}$ is convergent I wanted to use the alternate series test but I don't think i can use that because the series dosn't alternate every term any idea of what am I missing here?

Answer 1

Split the sum into two sums by $n\to 2n$ and $n\to 2n-1$ but consider only the first $Q$ terms

$\sum_{n=1}^Q\frac{(-1)^\frac{n(n-1)}{2}}{\sqrt n}=\sum_{n=1}^{Q/2}\frac{(-1)^n}{\sqrt {2n}}- \sum_{n=1}^{Q/2}\frac{(-1)^n}{\sqrt {2n-1}}$

$|{(-1)^n}/{\sqrt {2n-1}}|=1/{\sqrt {2n-1}}$ converge to $0$, and the two sums on the RHS are alternating. Hence the limit $Q\to\infty$ exist on the left hand side, and by equality for all $Q$ the limit also exist on the right hand side, so the sum in question is convergent. (and the limits are equal)

Answer 2

You're not missing anything, but you have powerful theorems to use, for example Dirichlet's Test . Can you spot what are $\;a_n,\,b_n\;$ here and whether the conditions are fulfilled?