# Infinite Series: sigma n^2/(n^2 +1)

by anderma8
Tags: infinite, n2 or n2, series, sigma
 Sci Advisor HW Helper PF Gold P: 4,771 You're again confusing the limit of the argument (here n^2/(n^2 +1)) with the actual sum, which is the limit of $$\sum_{n=1}^N\frac{n^2}{n^2+1}$$ as N-->infty. The theorem is saying that if the limit of the argument is not 0, then you must conclude that the sum diverges. If it IS 0, then you cannot conclude anything: the sum could converge or diverge. For instance, consider the old harmonic series $$\sum\frac{1}{n}$$ Sure, 1/n-->0 but it is well known that the harmonic series diverges nonetheless.