# Divergence of a reciprocal

Question: Say that $$\sum a_n$$ converges and $$a_n$$ does not equal 0 for all natural numbers.

Prove: $$\sum 1/a_n$$ diverges.

My understanding: I see that $$1/a_n$$ is the reciprocal of $$a_n$$. I have done many examples with numbers that prove this theorem, but I am not sure how to really prove this without testing numbers with convergence/divergence tests. Would I use the comparison theorem test here?

• $\lim a_n = ?$ And knowing this, $\lim \frac{1}{an} = ?$ – Thiago Nascimento Jul 30 '17 at 22:22
• – Martin Sleziak Dec 23 '19 at 10:51

If the series $\sum a_n$ converges, then the general term $a_n$ must go to zero when $n\to+\infty$.
What about $$\lim_{n\to +\infty}\frac {1}{a_n}$$