Sequences and Series – Convergence and Divergence of Series with Reciprocal Terms

Question

I am currently studying series and their convergence properties in my calculus course. I came across a problem that I've been trying to solve but haven't been able to crack yet.

Problem Statement:

Suppose that $$\sum_{n=0}^{\infty}a_n$$ is a series of positive terms which is convergent. Show that $$\sum_{n=0}^{\infty}\left(\frac{1}{a_n}\right)$$ is divergent. What about the converse?

My Attempt:

I understand that if a series $$\sum_{n=0}^{\infty}a_n$$ is convergent, then the sequence of its terms {a_n} must approach zero as n approaches infinity. However, I'm not sure how to apply this to the reciprocal series $$\sum_{n=0}^{\infty}\left(\frac{1}{a_n}\right)$$.

For the converse, intuitively it seems that if the reciprocal series is convergent, then the original series should be divergent. But I'm not sure how to prove this formally.

Background:

I am familiar with the basic tests for convergence (like the comparison test, root test, and ratio test), but I'm not sure how to apply them in this case. I would appreciate if the answer could be explained using these or similar concepts.

Thank you in advance for your help!

Answer 1

Convergence of $\sum_{n = 0}^\infty a_n$ implies that $a_n \to 0$. Hence $1/a_n \to \infty$ so that $\sum_{n = 0}^\infty 1/a_n$ cannot converge.

The converse

If $\sum_{n = 0}^\infty 1/a_n$ diverges, then $\sum_{n = 0}^\infty a_n$ converges

is not true. As a counterexample take $a_n = 1$ for all $n$.