> For the complete documentation index, see [llms.txt](https://coldmoon.gitbook.io/mathematical-analysis/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://coldmoon.gitbook.io/mathematical-analysis/mathematical-analysis-1/1.-limits.md).

# 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&#x20;

$$
\lim *{n\to \infty }{\frac {a*{n}}{b\_{n}}}=l
$$

### Theorem 3. Toeplitz limit theorem

Supports that $$n,k\subseteq \mathbb N^{+}$$,$$t\_{nk}\geq0$$and&#x20;

$$
\sum\_{k=1}^{n}{t\_{nk}} = 1,\quad \lim\_{n \rightarrow \infty}{t\_{nk}} = 0
$$

if $$\lim\_{n \rightarrow \infty}{a\_{n}} = a$$ , let $$x\_{n} = \sum\_{k=1}^{n}{t\_{nk}a\_{k}}$$, s.t.&#x20;

$$
\lim\_{n \rightarrow \infty}{x\_{n}} = a
$$

> 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:

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

$$\blacksquare$$