Show that $S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}$ converges for $n\in\Bbb N,\ x \in\Bbb R$ without using Taylor series.

Question

Given a sequence $\{S_n\}$, $n\in\Bbb N$:
$$
S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}
$$
Prove that $S_n$ converges for all $x\in\Bbb R$.

Please note that i know $S_n$ is a simple Taylor series for $e^x$, the case is I'm not supposed to know (and/or use) that fact when solving this problem since derivatives have not been defined yet.

I've started with a simpler case assuming $x = 1$. So the sequence becomes:
$$
S_n = 1+{1\over1!}+{1\over2!}+\cdots+{1\over n!}
$$

It seems reasonable to use Cauchy Criterion for the sequence. Namely suppose $m>n$, then we want to show:
$$
|x_n – x_m| < \epsilon \\
|x_m – x_n| = \left|\sum_{k=n+1}^m {1\over k!}\right|
$$
Lets try to synthetically bound the sum by:
$$
\begin{align}
{1\over k!} &= {1\over k!}\left(1 – {1\over k} + {1\over k}\right) \\
&\le {1\over k!}\left(1 + {1\over k-1} – {1\over k}\right) \\
&= {1\over k!}\left({k\over k -1} – {1\over k}\right) \\
&= {1\over (k – 1)(k-1)!} – {1\over k\cdot k!}
\end{align}
$$

By telescoping we obtain:
$$
\left|\sum_{k=n+1}^m {1\over k!}\right| \le \left|{1\over n\cdot n!} – {1\over m\cdot m!}\right|
$$

Given $m>n$ and $n,m \in \Bbb N$:
$$
\left|{1\over n\cdot n!} – {1\over m\cdot m!}\right| \le \left|1\over n\cdot n!\right| = {1 \over n\cdot n!}
$$

Applying the limit to $|x_m – x_n|$ one may obtain:
$$
0 \le \lim_{n\to\infty}|x_{n+p} – x_n| \le \lim_{n\to\infty} {1\over n\cdot n!} = 0
$$

So squeezing $|x_m – x_n|$ gives:
$$
\lim_{n\to\infty}|x_{n+p} – x_n| = 0,\ p\in \Bbb N
$$

Which would eventually mean:
$$
|x_m – x_n| < \epsilon
$$

However I'm not sure how to find the index $N_\epsilon$ from which the inequality becomes true since the expression for the upper bound involves a factorial.

If we now put $x = x_0 \in \Bbb R$:
$$
0 \le \lim_{n\to\infty}|x_{n+p} – x_n| \le \lim_{n\to\infty} {x_0\over n\cdot n!} = 0
$$
Which doesn't influence the value of the limit.

There are three questions in my mind:

1. Is the overall reasoning valid?
2. Is it possible to find a closed form of $N(\epsilon)$, such that $n, m > N_\epsilon \implies |x_n – x_m| < \epsilon$?
3. Should I consider two cases for $x\ge 0$ and $x<0$

Thank you!

Answer 1

You can simply use ratio test to show that the given series converges. Any sequence $S$ is converging if $$r=\lim_{n\to \infty}\frac{a_{n+1}}{a_n}<1$$ For this problem, $$r=\frac{x^{n+1}}{(n+1)!}\frac{n!}{x^n}$$ $$=\frac{x}{n+1}$$ $r$ goes to $0$ as $n\to\infty$.