Absolute convergence to a rational number Let's recall the not so popular/familiar form of completeness of real numbers:

Theorem: Absolute convergence of a series implies its convergence.

Since $\mathbb{Q} $ is not complete there should exist a series $\sum_{n=1}^{\infty} u_n$ with rational terms such that $\sum_{n=1}^{\infty} |u_n|$ converges to a rational number and $\sum_{n=1}^{\infty}u_n$ converges to an irrational number.

I could not think of an obvious example of such a series. Please provide one such example.


Every irrational number in Balanced ternary has a non-repeating expansion, and viceversa. Therefore, if we take a non-repeating sequence $(e_n)_{n\in\mathbb Z^+}$ with $e_n\in\{-1,1\}$, $$\sum_{k=1}^\infty\frac{e_k}{3^k}$$ will be irrational, while $$\sum_{k=1}^\infty\left|\frac{e_k}{3^k}\right|= \sum_{k=1}^\infty\frac1{3^k}=\frac12$$ will be rational.
