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When studying the material of convergent series, I came up with a question.

Is $\lim_{n \to \infty}a_n \ne 0$ a sufficient and necessary condition for the series $∑a_n$ to be divergent?

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4 Answers 4

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If $a_n$ does not go to $0$ as $n\to\infty$, then the sum will not converge.

However, even if it goes to $0$, the sum could still diverge, say $a_n=\cfrac{1}{n}$.

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It is sufficient, but not necessary, as $\lim_{n\to\infty}\frac{1}{n}=0$ but this gives the well-known harmonic series $$\sum\frac{1}{n}.$$We can check this diverges, for example, by the integral test.

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It is sufficient, but not necessary. For example, the harmonic series $$\sum \frac{1}{n}$$ diverges, even though $$\lim_{n \to \infty} \frac{1}{n} = 0.$$

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not necessary since $\sum\frac{1}{n}$ diverges, it is sufficient though as if $\sum\frac{1}{n}$ converges, then $a_n=S_n-S_{n-1}$ converges to $0$ (here $S_n$ is the partial sum $a_1+\cdots +a_n$).

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