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I'm interested in finding conditions about the convergence of a series. I have an increasing succession $\{a_n\}$.

The elements of $\{a_n\}$ are positive real numbers. I want to know the conditions where $$\sum_{n\gt1}^{\infty}\frac{1}{a_1a_2...a_n}$$ converge. I used the ratio test where i found that the series converges if $$\lim_{n \to \infty} \frac{1}{a_{n+1}}\lt1$$ Are there are more conditions?

I would appreciate additional insight, or hints.

Prime
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Daniel Vargas
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  • I don't understand why this is down-voted. The OP has asked a legitimate question and explained what he has done so far. Furthermore, what he did was correct. How has he violated any condition of the site? – saulspatz Jun 13 '18 at 01:27
  • @saulspatz. I agree with you................... – DanielWainfleet Jun 13 '18 at 02:04

1 Answers1

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If all of the $a_n$ are $\le 1$ then the $n$th term doesn't go to $0$ and the series diverges. Therefore, the condition you have found is necessary and sufficient.

saulspatz
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