# Is it true: $\sum_{n=1}^\infty \frac{a_n}{b_n}$&$\sum_{n=1}^\infty \frac{b_n}{c_n}$ are irrational=>$\sum_{n=1}^\infty \frac{a_n}{c_n}$ is irrational

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.

• The $a_n$, $b_n$ and $c_n$ are integers. – Gevorg Hmayakyan Jun 19 '17 at 6:04
• Could you explain what led you to conjecture such a result? – Did Jun 19 '17 at 12:07
• 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. – Gevorg Hmayakyan Jun 19 '17 at 12:41
• 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. – Did Jun 19 '17 at 13:58

Let $a_n = n+1, b_n = (-1)^n n(n+1), c_n = n(n+1)^2$

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

• Amazing. Interesting to see that the sum of this three sums are 0. – Gevorg Hmayakyan Jun 19 '17 at 6:20
• 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. – pisco Jun 19 '17 at 6:28

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

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

• This is more non trivial example. – Gevorg Hmayakyan Jun 19 '17 at 6:25
• Very ingenious! I tried to construct positive examples like these but didn't come up with any. – pisco Jun 19 '17 at 6:26
• 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. – Sangchul Lee Jun 19 '17 at 6:33
• 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. – Gevorg Hmayakyan Jun 19 '17 at 6:48