How can one check if a Cauchy-sequence converges in the rationals?

Question

Let $(x_k)$ be a sequence in $\mathbb Q$ such that $x_k=\sum\limits_{n=1}^{k}\frac{1}{10^{n^2}}$ for all $k\geq 1$.

It can be easily seen that this sequence is bounded and Cauchy. But does it converge in $\mathbb Q$? I could not find any way to verify that. Please help!

Answer 1

Note and recall:

• As you know the sequence converges in the reals, the sequence converges in rationals if an only if its limit is a rational number.

• The way the sequence is given you immediately get the decimal expansion of its limit.

• A rational number has an eventually periodic (or finite) decimal expansion.

So you need to check if the limit fulfills that last property.