<s>[INST] Proving that $0I need to prove that $0<e-S_{n}<\frac{4}{(n+1)!}$ $\quad\forall n\in\mathbb{N}\quad$ for $\quad S_{n}=1+1+\frac{1}{2!}+\ldots+\frac{1}{n!}$
Previously, I proved that $2<e<4$, and that $S_n$ is taylor polynomial of $e$.
I thought on proving it using induction:
Base: $S_1=1<2<e<4$
Step - for $n+1$:
$e-S_{n+1}=e-S_{n}-\frac{1}{(n+1)!}$
What am I missing here? It's supposed to be simple.
Thank you!
 [/INST] Let's consider inequality
$$x_n = \left( 1+ \frac{1}{n} \right)^n > 1+ \frac{n}{n} + \frac{n(n-1)}{2!} \frac{1}{n^2} + \cdots + \frac{n(n-1) \dots (n-k+1) }{k!} \frac{1}{n^k}$$
If we now consider limit $n \to \infty$, then we will have
$$e \ge 1+1 +\frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{k!} = S_k$$
Also we have
$$x_n =\left( 1+ \frac{1}{n} \right)^n < 1+1 +\frac{1}{2!} + \frac{1}{3!} + \cdots + \frac{1}{n!} = S_n$$
so, there is
$$x_n < S_n < e$$
which, after taking limit, finishes proof.
For estimation approximation let's consider
$$S_{n+k}-S_n= \frac{1}{(n+1)!}+ \cdots + \frac{1}{(n+k)!}< \frac{1}{(n+1)!}\left( \frac{1}{(n+2)} + \frac{1}{(n+2)^2} +\cdots \right) = \\= \frac{1}{(n+1)!} \frac{n+2}{(n+1)}< \frac{1}{n \cdot n!}$$
Fixing $n$ and taking limit $k \to \infty$ gives
$$0<e-S_n<\frac{1}{n \cdot n!}<\frac{4}{(n+1)!}$$
</s>