If $\sum\frac1{a_n}$ is convergent, then irrational?

$\{a_n\}$ is a strictly increasing sequence of positive integers such that $$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=1$$ If $\sum\limits_{n=1}^\infty\frac1{a_n}$ is convergent, can one conclude that $\sum\limits_{n=1}^\infty\frac1{a_n}$ is an irrational number? a transcendental number?

A special case is $\zeta(n)(n\geq2)$. so, the question, if true, may be difficult.

Does someone suggest a counter-example? Thanks a lot!

Take $a_n = n(n+1)$. Then

$$\sum_{n=1}^\infty \frac{1}{a_n} = \sum_{n=1}^\infty \left(\frac{1}{n} - \frac{1}{n+1}\right) = 1.$$

Define $a_n=n(n+1)$. Then $$\lim_{n\to\infty} \frac{a_{n+1}}{a_n}=\lim_{n\to\infty} \frac{n^2+3n+2}{n^2+n}=1$$

And $$\sum_{n=1}^\infty\frac1{a_n}=\lim_{n\to\infty}\sum_{k=1}^n\left(\frac 1k-\frac1{k+1}\right)=\lim_{n\to\infty}\left(1-\frac 1{n+1}\right)=1\in\Bbb Q$$