Does that series converge or diverge? Does the series $$\sum \limits _{n=3}^\infty \frac{(-1)^{[\log n]}}{\sqrt{n}}$$ converge or diverge? As usually, $[x]$ denotes the integer part of $x.$

Calculate the sum, $n=\lceil e^k\rceil$ to $\lfloor e^{k+1}\rfloor$. There are about $(e-1)e^k$ terms. Each has the same sign, and each has absolute value $\ge \frac{1}{\sqrt{e^{k+1}}}$. So the Cauchy Criterion for the partial sums fails, and the series does not converge.