# The sum of the series $\sum_{n=0}^{\infty}\frac{\epsilon_n}{n!}$ is an irrational number

Let $\{\epsilon_n\}$ be a sequence where $\epsilon_n$ is either $1$ or $-1$. How could I Show
that the sum of the series

$$\sum_{n=0}^{\infty}\frac{\epsilon_n}{n!}$$

is an irrational number.

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Just repeat the arguments in the regular proof that e is irrational. – achille hui May 9 '13 at 15:17
As a matter of fact, using the proof/arguments listed below, we can generalize the result further. As long as $\epsilon_n \in \{-1,1\}$ comes in infinite quantities, you can allow some (even if infinite) $\epsilon_n$ be 0. That is, $\epsilon_n \in \{-1,0,1\}$. You simply cannot have a finite number of both $-1,1$. – CogitoErgoCogitoSum Nov 29 '14 at 1:16

If there exist $p\in\mathbb{Z}$ and $q\in\mathbb{N}$, such that $\frac{p}{q}=\sum_{n=0}^\infty\frac{\epsilon_n}{n!}$, then $q!\cdot\sum_{n=q+1}^\infty\frac{\epsilon_n}{n!}$ must be an integer. However, $$|q!\cdot\sum_{n=q+1}^\infty\frac{\epsilon_n}{n!}-\frac{\epsilon_{q+1}}{q+1}|\le\sum_{n=q+2}\frac{q!}{n!}<\frac{1}{(q+1)(q+2)}\cdot\sum_{m=0}\frac{1}{2^m}=\frac{2}{(q+1)(q+2)},$$ which implies that $$0<\frac{1}{q+1}-\frac{2}{(q+1)(q+2)}\le |q!\cdot\sum_{n=q+1}^\infty\frac{\epsilon_n}{n!}|\le \frac{1}{q+1}+\frac{2}{(q+1)(q+2)}<1,$$ a contradiction.
The proof is by contradiction. Define: $$S = \sum_{k \ge 0} \frac{\epsilon_k}{n!}$$ Assume $S$ is rational, i.e. there are $u \in \mathbb{Z}$, $v \in \mathbb{N}$ such that $S = u / v$.
Pick $b > v$, so that $b \ge 2$. Then $b! S$ is an integer, i.e.: $$S = \sum_{0 \le k \le b} \frac{b! \epsilon_k}{k!} + \sum_{k > b} \frac{b! \epsilon_k}{k!}$$ The first sum is an integer, so the second sum has to be an integer too. Now: $$\frac{b! \epsilon_k}{k!} = \frac{\epsilon_k}{(b + 1) (b + 2) \ldots k}$$ But: $$\frac{1}{(b + 1) (b + 2) \ldots k} < \frac{1}{b^{k - b}}$$ By the triangular inequality: $$\left\rvert \sum_{k \ge b + 1} \frac{b! \epsilon_k}{k!} \right\rvert \le \sum_{k \ge b + 1} \frac{b!}{k!} < \sum_{k \ge b + 1} b^{-k} = b^{- b - 1} \sum_{k \ge 0} b^{-k} = b^{- b - 1} \frac{1}{1 - 1 / b} = \frac{1}{b^{b +1} (b - 1)} < \frac{1}{b^{b + 1}}$$ So: $$1 = \left\lvert \frac{b! \epsilon_b}{b!} \right\rvert > \left\rvert \sum_{k \ge b + 1} \frac{b! \epsilon_k}{k!} \right\rvert$$ and the "leftover sum" can never be 0, so it isn't an integer.