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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?

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hint

If the series $\sum a_n $ converges, then the general term $a_n $ must go to zero when $n\to+\infty $.

What about $$\lim_{n\to +\infty}\frac {1}{a_n} $$

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\begin{align*}\sum_{n=1}^\infty a_n\text{ converges}&\Longrightarrow\lim_{n\in\mathbb N}a_n=0\\&\Longrightarrow\lim_{n\in\mathbb N}\frac1{|a_n|}=+\infty\\&\Longrightarrow\sum_{n=1}^\infty\frac1{a_n}\text{ diverges.}\end{align*}

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