How to show that $e$ is irrational by studying $\sum^{n}_{k=1}\frac{1}{k!}$ and $\sum^{n}_{k=0}\frac{1}{k} + \frac{1}{n \cdot n!}$?
63960
proving "e" irrational using convergent series
97684
Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$
107789
Sum of $\sum \limits_{n=0}^{\infty} \frac{1}{(kn)!}$
150642
How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?
76296
Compute $\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{n^{k}}{n!}$ for $k=0, 1, 2 ... $
64336
Show $\sum_{k=1}^n \frac{1}{k^2} \le 2$ and $\ln(n!) \ge 1 -n+n\ln(n)$ for all positive integers n
13592
How can $\pi$ be irrational if it can be calculated using an infinite series of rational numbers?
83313
Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$
16143
How can I prove that $\sum_{n=1}^\infty \frac{1}{n(n+1)} = 1$?
118195
How does one prove that $e$ exists?
40009
Show that $e^{\sqrt 2}$ is irrational
87596
Is $\sum_{p}\frac{1}{p^{2}}$ irrational?
62884
Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number
80216
Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$
139229
How to prove by arithmetical means that $\sum\limits_{k=1}^\infty \frac{((k-1)!)^2}{(2k)!} =\frac{1}{3}\sum\limits_{k=1}^{\infty}\frac{1}{k^{2}}$
21791
Show that $\sum\limits_{k=1}^\infty \frac{1}{k(k+1)(k+2)\cdots (k+p)}=\frac{1}{p!p} $ for every positive integer $p$
111524
Information on the sum $\sum_{n=1}^\infty \frac{\log n}{n!}$