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• Notes

هي مجموعة أرقام مُرتبة بترتيب معين، ونرمز للأرقام بـ a1 و a2 و a3 الى an حيث :

: الحد الأول ــ  الحد الثانى

وبشكل عام ،  هو الحد النونى أيا كانت قيمه  .

The sequence {a1, a2, a3, . . .} is also denoted by  an

Definition A sequence $\left\{a_{n}\right\}$ has the limit $L$ and we write

$\lim _{n \rightarrow \infty} a_{n}=L$ or $\quad a_{n} \rightarrow L$ as $n \rightarrow \infty$

if we can make the terms $a_{n}$ as close to $L$ as we like by taking $n$ sufficiently large.

If $\lim_{n \rightarrow \infty} a_{n}$ exists, we say the sequence converges (or is convergent). Otherwisewe say the sequence diverges (or is divergent).

نظرية هامة جدا:

If $\lim _{x \rightarrow \infty} f(x)=L$ and $f(n)=a_{n}$ when $n$ is an integer, then $\lim _{n \rightarrow \infty} a_{n}=L$

Limit Laws for Sequences

If $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are convergent sequences and $c$ is a constant, then

$\lim _{n \rightarrow \infty}\left(a_{n}+b_{n}\right)=\lim _{n \rightarrow \infty} a_{n}+\lim _{n \rightarrow \infty} b_{n}$

$\lim _{n \rightarrow \infty}\left(a_{n}-b_{n}\right)=\lim _{n \rightarrow \infty} a_{n}-\lim _{n \rightarrow \infty} b_{n}$

$\lim _{n \rightarrow \infty} c a_{n}=c \lim _{n \rightarrow \infty} a_{n} \quad \lim _{n \rightarrow \infty} c=c$

$\lim _{n \rightarrow \infty}\left(a_{n} b_{n}\right)=\lim _{n \rightarrow \infty} a_{n} \cdot \lim _{n \rightarrow \infty} b_{n}$

$\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}=\frac{\lim _{n \rightarrow \infty} a_{n}}{\lim _{n \rightarrow \infty} b_{n}}$ if $\lim _{n \rightarrow \infty} b_{n} \neq 0$

$\lim _{n \rightarrow \infty} a_{n}^{p}=\left[\lim _{n \rightarrow \infty} a_{n}\right]^{p}$ if $p>0$ and $a_{n}>0$

Squeeze Theorem for Sequences

If $a_{n} \leq b_{n} \leq c_{n}$ for $n \geq n_{0}$ and $\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} c_{n}=L,$ then $\lim _{n \rightarrow \infty} b_{n}=L$

Theorem  If $\lim _{n \rightarrow \infty}\left|a_{n}\right|=0,$ then $\lim _{n \rightarrow \infty} a_{n}=0$

تعريفات هامة جدا:

The sequence $\left\{r^{n}\right\}$ is convergent if $-1<r \leq 1$ and divergent for all other

values of $r .$

$\lim _{n \rightarrow \infty} r^{n}=\left\{\begin{array}{ll}{0} & {\text { if }-1<r<1} \\ {1} & {\text { if } r=1}\end{array}\right.$

Definition A sequence $\left\{a_{n}\right\}$ is called increasing if $a_{n}<a_{n+1}$ for all $n \geq 1$

that is, $a_{1}<a_{2}<a_{3}<\cdots$ It is called decreasing if $a_{n}>a_{n+1}$ for all $n \geq 1$

A sequence is monotonic if it is either increasing or decreasing.

Definition A sequence $\left\{a_{n}\right\}$ is bounded above if there is a number $M$

such that

$a_{n} \leq M \quad$ for all $n \geq 1$

It is bounded below if there is a number $m$ such that

$m \leq a_{n} \quad$ for all $n \geq 1$

If it is bounded above and below, then $\left\{a_{n}\right\}$ is a bounded sequence.

Monotonic Sequence Theorem Every bounded, monotonic sequence is

convergent.