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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!

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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.

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  • $\begingroup$ Note that a finite decimal expansion is eventually periodic (with a bunch of zeros), so your statement (or finite) is redundant ;) +1! $\endgroup$ Jun 2, 2015 at 12:19
  • $\begingroup$ I agree, and this is why I put in in parenthesis. But I still meant to mention it so that it is not overlooked. Possibly I should have phrased this more clearly. $\endgroup$
    – quid
    Jun 2, 2015 at 12:34
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    $\begingroup$ it was clear. My comment was just a note for the others :D $\endgroup$ Jun 2, 2015 at 12:46

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