I am supposed to show that it converges or diverges. If it diverges, I need to further demonstrate whether it converges absolutely or conditionally.
Here is the problem I am having trouble with so far that I happened to get right (through the use of my logic but I don't know how to write it in math terms - I'll explain in detail as you read):
Problem: sum of ((n^2 + 1)/(2n^2 + 1))^n from 1 to inf - Wolfram|Alpha
My logic is that the limit as n->inf of (n^2 + 1)/(2n^2 + 1) is 1/2 and (1/2)^n is a geometric series and it therefore should converge but this seems to be not good enough to me because of the fact that the first term for isntance is 2/3 and the second one is 5/9 which are somewhat away from 1/2 but then on the other hand the part of my logic that says it is still okay is that even the first term even though it's not near 1/2 is less than 1 which, in a geometric series, implies convergence.
Any input would be GREATLY appreciated!
Thanks in advance!