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Assume that the: $$\sum_{n=1}^\infty \frac{a_n}{b_n}$$ and $$\sum_{n=1}^\infty \frac{b_n}{c_n}$$ are convergent and irrational, then if $$\sum_{n=1}^\infty \frac{a_n}{c_n}$$ is convergent should it also be irrational then for the integer $a_n$, $b_n$ and $c_n$?

I assume this is false, but can not find any counterexample.

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    $\begingroup$ The $a_n$, $b_n$ and $c_n$ are integers. $\endgroup$ – Gevorg Hmayakyan Jun 19 '17 at 6:04
  • $\begingroup$ Could you explain what led you to conjecture such a result? $\endgroup$ – Did Jun 19 '17 at 12:07
  • $\begingroup$ If we consider the product of first and second sums if they are irrational this will give us some sum set of coefficients $\sum\frac{a_n}{b_n}\frac{b_{n-k}}{c_{n-k}}$. In the most of cases this will bring us to another irrational value except the cases with $\frac{1}{a}*a$. So I wanted to understand if the form of the sum is critical for the irrationality. So I started from the simplest case. $\endgroup$ – Gevorg Hmayakyan Jun 19 '17 at 12:41
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    $\begingroup$ You seem to be confusing the fact that the values of a sequence are rational, resp. irrational, with the fact that their sum is rational, resp. irrational. Each of these four cases may happen, as simple examples show. $\endgroup$ – Did Jun 19 '17 at 13:58
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Let $a_n = n+1, b_n = (-1)^n n(n+1), c_n = n(n+1)^2$

Then $$\sum_{n=1}^{\infty} \frac{a_n}{b_n} = -\ln 2$$ $$\sum_{n=1}^{\infty} \frac{b_n}{c_n} = \ln 2 -1$$ $$\sum_{n=1}^{\infty} \frac{a_n}{c_n} = 1$$

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  • $\begingroup$ Amazing. Interesting to see that the sum of this three sums are 0. $\endgroup$ – Gevorg Hmayakyan Jun 19 '17 at 6:20
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    $\begingroup$ These three sums give 0 is rather a coincidence I think, if you multiply $b_n$ by an arbitrary integer, then the three sums no longer sum to 0. $\endgroup$ – pisco Jun 19 '17 at 6:28
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Another example with all terms positive is

$$ a_n = 2n+1, \qquad b_n = n(n+1)(2n-1), \qquad c_n = n(n+1)(n+2)(n+3)(2n+1). $$

Then we can prove that

$$ \sum_{n=1}^{\infty} \frac{a_n}{b_n} = \frac{1}{3} + \frac{8}{3}\log 2, \qquad \sum_{n=1}^{\infty} \frac{b_n}{c_n} = \frac{9}{15} - \frac{8}{15}\log 2, \qquad \sum_{n=1}^{\infty} \frac{a_n}{c_n} = \frac{1}{18}. $$

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  • $\begingroup$ This is more non trivial example. $\endgroup$ – Gevorg Hmayakyan Jun 19 '17 at 6:25
  • $\begingroup$ Very ingenious! I tried to construct positive examples like these but didn't come up with any. $\endgroup$ – pisco Jun 19 '17 at 6:26
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    $\begingroup$ Thank you all for your comments. Since $\sum_{n=1}^{\infty} R(n)$ with a rational function $R(x)$ usually spits out polygamma functions, we have a rather large room for choosing sequences subject to $a_n/c_n$ having a special form. $\endgroup$ – Sangchul Lee Jun 19 '17 at 6:33
  • $\begingroup$ Is it possible to find an example with different irrationality in the first and second sums. I mean the $\log 2$ and $\pi$ for example. $\endgroup$ – Gevorg Hmayakyan Jun 19 '17 at 6:48

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