Advanced Calculus Infinite Series

[jtn2007]

Homework Statement

Suppose that {a_n} is a monotone decreasing sequence of positive numbers. Show that if the series an converges, then the lim(nan)=0.

Homework Equations

The Attempt at a Solution

I started the proof with the fact that since I know the sequence is monotone decreasing and the the series converges then the lim an=0 but I am not sure how to show that the lim nan=o.

any help would be greatly appreciated

 

[Dick]

It's really a lot like the proof the harmonic series diverges. If n*a_n does not converge to 0, then there is a positive constant e such that n*a_n>e for an infinite number of n, right? Use that with the monotonicity to show a_n must diverge.