1. Limits

Definition: series limit

$$\left(\lim _{n \rightarrow \infty} x_{n}=A\right):=\forall \varepsilon>0\quad \exists N \in \mathbb{N}\quad \forall n>N\quad \left(\left|x_{n}-A\right|<\varepsilon\right)$$

We say that the sequence $\{x_n\}$ converges to $A$ or tends to $A$ and write $x_n \to A$ as $n \to \infty$.

Definition: fundamental or Cauchy sequence

A sequence $\{x_n\}$ is called a fundamental or Cauchy sequence if for any $\varepsilon > 0$ there exists an index $N \in N$ such that $|x_m − x_n| < \varepsilon$ whenever $n > N$ and $m > N$.

Theorem: Weierstrass

In order for a nondecreasing sequence to have a limit, it is necessary and sufficient that it be bounded above.

Two important limit

$$\text { e }:=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}$$

$$\lim_{x\to 0}\frac{\sin x}{x}=1$$

Definition 3. inferior limit and superior limit

$$\varliminf_{k \rightarrow \infty} x_{k}:=\lim _{n \rightarrow \infty} \inf _{k \geq n} x_{k}$$

$$\varlimsup_{k \rightarrow \infty} x_{k}:=\lim _{n \rightarrow \infty} \sup _{k \geq n} x_{k}$$

Theorem 2. Stolz

Let ${\displaystyle (a{n}){n\geq 1}}$ and ${\displaystyle (b{n}){n\geq 1}}$ be two sequences of real numbers. Assume that ${\displaystyle (b{n}){n\geq 1}}$ is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching${\displaystyle +\infty }$ , or strictly decreasing and approaching $-\infty$) and the following limit exists:

$$\lim_ {n\to \infty }{\frac {a_{n+1}-a_{n}}{b_{n+1}-b_{n}}}=l$$

Then, the limit

$$\lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}=l$$

Theorem 3. Toeplitz limit theorem

Supports that $n,k\subseteq \mathbb N^{+}$,$t_{nk}\geq0$and

$$\sum_{k=1}^{n}{t_{nk}} = 1,\quad \lim_{n \rightarrow \infty}{t_{nk}} = 0$$

if $\lim_{n \rightarrow \infty}{a_{n}} = a$ , let $x_{n} = \sum_{k=1}^{n}{t_{nk}a_{k}}$, s.t.

$$\lim_{n \rightarrow \infty}{x_{n}} = a$$

By using $t_{nk}=\frac{1}{n}$, we can quickly infer The Cauchy proposition theorem.

By using $t_{n k}=\frac{b_{k+1}-b_{k}}{b_{n+1}-b_{1}}$, we can quickly infer The Stolz theorem.

Stirling's formula

Specifying the constant in the $\mathcal O(\ln n)$ error term gives $\frac12 \ln(2\pi n)$, yielding the more precise formula:

$$n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}$$

$\blacksquare$