Prove that ∑ 1/a_n diverges if ∑ a_n converges

by RoRi

March 17, 2016

Prove that if $a_n > 0$ for all $n \in \mathbb{Z}$ and if $\sum_n a_n$ converges, then $\sum_n \frac{1}{a_n}$ diverges.

Proof. Since $\sum a_n$ converges we know $\lim_{n \to \infty} a_n =0$ . By the definition limit this means that for all $\varepsilon > 0$ there exists an integer $N$ such that

$|a_n| < \varepsilon \qquad \text{whenever} \qquad n \geq N.$

In particular, if we take $\varepsilon = 1$ then we know there is some $N$ such that $n \geq N$ implies $|a_n| < 1$ . But then,

$|a_n| < 1 \quad \implies \quad \left| \frac{1}{a_n} \right| > 1 \quad \implies \quad \lim_{n \to \infty} \frac{1}{a_n} \neq 0.$

Hence, the series $\sum \frac{1}{a_n}$ diverges $. \qquad \blacksquare$

Stumbling Robot

Prove that ∑ 1/a_n diverges if ∑ a_n converges

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