Honors Calculus I - 2.1 Sequences

Sequences

An infinite sequence $\{x_n\}$ is converge to $x$ ($x$ is called the limit)iff:

$$\forall \varepsilon > 0,\;\exist N\;\;\text{that}\;\forall n\ge N,|x_n-x|<\varepsilon$$

If not, it is called divergent.

Above is called the $\varepsilon -N$ definition. (which can also use to prove ${x_n}$ is converge to $N$)

Equivalent statement :

For every $\varepsilon^{*}>0$, there exists a number $N$ such that for all $n\ge N$, the equality $|x_n-x|<f(\varepsilon^{*})$ holds, where $f$ is a function satisfying $f(\varepsilon^{*})<\varepsilon$.

Proof Method / Example :

Little - o notation

Properties

• If $x_n\to x$, and $x_n\to x'$, Then $x=x'$
• Sequences $\{x_n\}$ converges to $x$ iff $x_n\in N_{\varepsilon}(x)$ for all but finitely many $n$
• If ${x_n}$ converges, then ${x_n}$ is bounded.

Arithmetic Rules

• Sum Rule, Difference Rule, Product Rule, Quotient Rule

the Order Rule

Suppose $\lim_{n\to +\infty}x_n=x,\lim_{n\to+\infty}y_n=y$

• If $x_n\le y_n\;\;\forall n\ge n_0$, then $x\le y$
• If $x<y$, then exists $n_0$ satisfy $n\ge n_0, x_n<y_n$

Squeeze Theorem for sequences

If $x_n\le y_n\le z_n$, and if $\lim_{n\to \infty}x_n=\lim_{n\to \infty}z_n=L$, then $\lim_{n\to \infty}y_n=L$

Several Common Limits

Monotonic Sequences

Monotone Convergence Theorem : A monotonic sequences converges iff it is bounded

A trick:

To prove a sequence is converge :

1. Construct two bounded monotonic sequences, then we know they are converge
2. Prove $\lim_{n\to \infty}(a_n-b_n)=0$, So the two sequences have the same limit
3. Squeeze Theorem

Euler’s number :

$$e=\lim_{n\to \infty}(1+\frac 1 n)^n$$

Subsequences

Omit some terms from a sequence $\{x_n\}$, and we get a subsequence $\{x_{n_k}\}$, which is also infinite.

• Convergence of subsequence : If $\{x_n\}$ converges to $x$, then every $\{x_{n_k}\}$ also converges to $x$
• Bolzano-Weierstrass Theorem : Every bounded sequence has at least one convergent subsequence. （这个东西很重要，用于解决自变量收敛的问题）

Cauchy Criterion

A sequence $\{x_n\}$converge iff :

$$\forall \varepsilon>0,\;\; \exist N>0,\;\; \forall n,m\ge N\\ |x_n-x_m|<\varepsilon$$

Infinite Limit of Sequence

$\{x_n\}$ is divergent to $\infty$ if :

$$\forall M\in \R\,\;\;\exist N>0\,\;\;\forall n\ge N\\ x_n\ge M$$

We can write $\lim_{n\to \infty}x_n=\infty$.

• If $\lim_{n\to \infty}x_n=\infty$, then $\lim_{n\to\infty}cx_n=\text{sign}(c)\infty$
• Suppose $x_n>0$ for all $x_n$. Then $\lim_{n\to \infty}x_n=\infty$ iff $\lim_{n\to\infty}\dfrac 1{x_n}=0$.

Note that :

If we use change variable to calculate limit, note the limit of the variable itself, for example:

$$\lim_{n\to \infty} n\sin (\frac 1 n)=\lim_{n\to 0}\frac 1 n\sin n$$