While I was studiyng I found some criterions to prove that an infinite convergent serie has an irrational sum.
That got me thinking. There are any criterions that prove the rationality of an infinite convergent serie sum?
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Potential duplicate: https://math.stackexchange.com/questions/66518/determine-if-sum-of-series-is-rational-or-not – symplectomorphic Mar 26 '18 at 15:26
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There are particular cases. For example for $n\in \Bbb N$ we have $n!e=n!\sum_{k=0}^{\infty}\frac {1}{k!}=A(n)+B(n)$ where $A(n)\in \Bbb N$ and $B(n)=\sum_{k=n+1}^{\infty}\frac {n!}{k!}\in (0,1).$ Therefore $n!e\not \in \Bbb N$ for all $n\in \Bbb N$, so $e$ is irrational. – DanielWainfleet Mar 27 '18 at 01:45