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Rational number

Prove that there exist infinite many positive real numbers x such that the sum

    \[\mathcal{S}(x)=\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n+x}\]

is rational.

Solution

It suffices to prove that \mathcal{S} is continuous. Indeed,

    \begin{align*} \left | \mathcal{S}(x) - \mathcal{S}(y) \right | &=\left | \sum_{n=1}^{\infty} (-1)^{n-1} \left ( \frac{1}{n+x} - \frac{1}{n+y} \right ) \right | \\ &\leq \sum_{n=1}^{\infty} \frac{\left | x-y \right |}{\left ( n+x \right )\left ( n+y \right )} \\ &\leq \left | x-y \right | \sum_{n=1}^{\infty} \frac{1}{n^2} \\ &\leq 4 \left | x-y \right | \end{align*}

Thus \mathcal{S} is continuous. Since \mathcal{S} is non constant , it follows that \mathcal{S}((0, +\infty)) is an interval. The result follows.

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