Series from 1 to ~ of (-1)^n * (n/10)^n converge or diverge?

Question

Pls give prove .
Thank you 4 your help

Answer 1

it diverges no matter what, because you can't know if the last number is positive or negative

in this case the 'last' one is infinite though, so the series will diverge anyway

(n/10)^n for n is infinite translates to infinite^infinte which is infinite of course

Answer 2

Yes this series diverges. Consider the fact that for n>10, n/10>1 and so as n→∞, so too must (n/10)^n. Since it is necessary (but not sufficient) for the series to converge that the limit as n→∞ of the individual terms is 0, and that is clearly not the case there, the series diverges.

Answer 3

The size of the terms is (n/10)^n which tends to infinity as n tends to infinity. Therefore, the series diverges (any convergent series must have terms that tend to 0).

Answer 4

The serie in absolute value is a_n = n^n/10^n which tends to infinity. (Orders of infinitud)
So the serie can not converge