I'd like to find for which values of x does
(1) sum n=1 to infinity (x^n)/n converge?
(2) sum n=1 to infinity (x^n)/(n^2) converge?
(3) sum n=1 to infinity (n^x)/(n^2 +1) converge?
Do i all do the ratio test and set the limit of it less than 1??? or is there different rules to consider for the each cases???
for instance for the (1) I took the ratio test and then I set the limit of it less than 1
lim n->oo (x^n+1)/(n+1) <1 which then becomes (x^n+1) has to be less than n+1
then x must be less than (n+1)^-(n+1) is this the right way of solving the problem??? help !!