Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 1 down vote accepted

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 $a_n=\cfrac{1}{n}$.

share|cite|improve this answer

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.

share|cite|improve this answer

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

share|cite|improve this answer

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 $a_1+\cdots +a_n$).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.