(Xn) = (1 + 1/n^2)^n^2
i) does it converge?
ii) find the limit.
attempt at a solution for i):
- prove that it is monotonic and bounded
=> convergence
- bounded:
|(Xn)| <= M , all n in Naturals
(1 + 1/n^2)^n^2 <= M
n^2ln((n^2 + 1)/n^2) <= lnM
n^2ln(n^2 + 1)-n^2ln(n^2) <= lnM
n^2ln(n^2 + 1)-n^2ln(n^2) <= n^2ln(n^2 + 1) so,
n^2ln(n^2 + 1) <= lnM
not sure how to continue...
ii) have yet to try
thanks