Question

How do you test the alternating series `Sigma (-1)^nsqrtn` from n is `[1,oo)` for convergence?

Answer 1

As `lim_(n->oo) sqrtn = +oo` the series does not satisfy Cauchy's necessary condition and thus cannot be convergent.

Besides let `a_n = (-1)^n sqrtn`.
Clearly `a_(2k) > 0` and `a_(2k+1) < 0` so the series oscillates indefinitely.