calculus
posted by Joe .
Use the ratio test to determine whether the series is convergent or divergent.
1/2 + 2^2/2^2 + 3^2/2^3 + 4^2/2^4 + ...

As n> infinity, the ratio of successive terms, [(n+1)/n]^2*(1/2), approaches 1/2
Read about the ratio test here (if you are not already familiar with it):
http://en.wikipedia.org/wiki/Ratio_test
That will tell you what the answer is in this case.