Can a Convergent Series Prove the Divergence of Another Series?

[chooooch]

Homework Statement

Suppose that ∑_(n=1)^∞▒a_n where a_n≠0 is known to be a convergent series. Prove that ∑_(n=1)^∞▒1/a_n is a divergent series.

Homework Equations

The Attempt at a Solution

if ∑_(n=1)^∞▒a_n is convergent then lim n→∞ a_n = 0. thus for some N a_n>1 and a_n >1/a_n if n>N. Thus, ∑_(n=1)^∞▒1/a_n diverges by comparison test.
I don't know if this is right, I don't know what to do.

 

[Dick]

A series can only converge if it's nth term approaches 0. You see this by looking at the limit of the partial sums. If limit a_n=0 then limit 1/a_n cannot be zero. As I think you were trying to say, for some N>0 and all n>N |a_n|<1. That would mean |1/a_n|>1.