Intuitively speaking, I first thought that if the series $\Sigma a_n$ is divergent then

$$\lim_{n \to \infty} a_n \ne 0$$

therefore it was clear that $\Sigma \frac{a_n}{1+a_n} $ would be divergent, but when I thought about it there are cases where the limit of the sequence does approach to $0$ and yet diverge, like the harmonic series.

Then I tried to go with since the sequence diverges, the series is not Cauchy (I m not even 100% sure if this is true but I tried)

$$|\sum_{i = m}^{n} a_n| \gt \epsilon$$

and derive the other series to not be Cauchy as well, only to not being able to reach.

I appreciate all the help.


marked as duplicate by Alex M., colormegone, Ramiro, Shailesh, choco_addicted May 2 '16 at 0:56

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  • $\begingroup$ Is this a problem from Rudin? If so, you should add that the terms are positive. $\endgroup$ – user1337 Jun 4 '13 at 19:30
  • $\begingroup$ @Panda Does this hold if the terms are not necessarily positive? $\endgroup$ – Potato Jun 4 '13 at 19:32
  • 5
    $\begingroup$ See this, with the assumption of positive $a_n$. $\endgroup$ – David Mitra Jun 4 '13 at 19:37
  • $\begingroup$ @Potato if negative values are to be allowed, it should at least say that no term is minus one. I don't know if the result remains valid. $\endgroup$ – user1337 Jun 4 '13 at 19:37
  • 1
    $\begingroup$ Please note that this is not a duplicate of the linked question, because here $a_n$ is not assumed to be positive. $\endgroup$ – 23rd Jun 4 '13 at 20:30
  1. If $(a_n)$ is a non-negative sequence, as commented by Panda and David Mitra, the statement is true. This is because, if $\sum_n\frac{a_n}{1+a_n}$ converges, then $\lim\limits_{n\to\infty} a_n=0$, so by comparison test, $\sum_n a_n$ converges, a contradiction.
  2. In general, the statement is false. For example, as a similar construction here, let $b_n=\frac{(-1)^n}{\sqrt{n}}$ and let $a_n=\frac{b_n}{1-b_n}$. Then $b_n=\frac{a_n}{1+a_n}$ and $$a_n=b_n+b_n^2+\frac{b_n^3}{1-b_n}.\tag{1}$$ It is easy to see that $\sum_n b_n$ is convergent, $\sum_n b_n^2$ is divergent and $\sum_n \frac{b_n^3}{1-b_n}$ is convergent( because $b_n\to 0$ and $\sum_n |b_n|^3$ is convergent), so from $(1)$ we know that $\sum_n a_n$ is divergent.
  • $\begingroup$ Nice counterexample. $\endgroup$ – Potato Jun 4 '13 at 20:17
  • $\begingroup$ It's great to have math geniuses. Thank you very much. $\endgroup$ – hyg17 Jun 5 '13 at 5:50
  • $\begingroup$ @hyg17: You are welcome! I have to clarify that I am definitely not a math genius. :) $\endgroup$ – 23rd Jun 5 '13 at 6:16

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