Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge?

Question

Does the series: $\sum_{n=1}^\infty (-1)^n \lbrack {\sqrt\frac{n}{2}} \rbrack$ Converge ?

Note: by the brackets I mean the floor function.

I tried to substitute numbers and look at the members of the sum, and I got something like this: $0, 1, -1, 1 , -2, 2 ........ -k,k...$ I don't think the sum of these numbers will lead any where, and the series diverges but I don't know to to prove this mathematically. any kind of help would be appreciable.

Answer 1

Given the series $$ \sum_{n=0}^{+\infty} a_n $$ the necessary condition for convergence is: $$ \lim_{n \to \infty} a_n = 0 $$ but that's not your case.

Showing this is enough to prove formally the divergence of the series.