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• Notes

هي عملية جمع لـ عدد لا نهائي من الحدود  a1+a2+a3….+an ويرمز لها بالرمز

$\sum_{n=1}^{\infty} a_{n} \quad$ or $\quad \sum a_{n}$

Definition Given a series $\sum_{n=1}^{\infty} a_{n}=a_{1}+a_{2}+a_{3}+\dots,$ let $s_{n}$ denote its

nth partial sum:

$s_{n}=\sum_{i=1}^{n} a_{i}=a_{1}+a_{2}+\dots+a_{n}$

If the sequence $\left\{s_{n}\right\}$ is convergent and $\lim _{n \rightarrow \infty} s_{n}=s$ exists as a real number, then

the series $\Sigma a_{n}$ is called convergent and we write

$a_{1}+a_{2}+\cdots+a_{n}+\cdots=s \quad$ or $\quad \sum_{n=1}^{\infty} a_{n}=s$

The number $s$ is called the sum of the series. If the sequence $\left\{s_{n}\right\}$ is divergent,

then the series is called divergent.

الـ geometric series

هي series  ليها ratio  ثابت بين مجموعة متتالية من الارقام

نظرية: اذا كانت  الـ series      (Convergent)   فيكون الـ Limit عندما تؤول n إلى مالانهاية، تساوي صفر.

Theorem If $\Sigma a_{n}$ and $\Sigma b_{n}$ are convergent series, then so are the series $\Sigma c a_{n}$

(where $c$ is a constant), $\Sigma\left(a_{n}+b_{n}\right),$ and $\Sigma\left(a_{n}-b_{n}\right),$ and

(i) $\sum_{n=1}^{\infty} c a_{n}=c \sum_{n=1}^{\infty} a_{n}$

(ii) $\sum_{n=1}^{\infty}\left(a_{n}+b_{n}\right)=\sum_{n=1}^{\infty} a_{n}+\sum_{n=1}^{\infty} b_{n}$

(iii) $\sum_{n=1}^{\infty}\left(a_{n}-b_{n}\right)=\sum_{n=1}^{\infty} a_{n}-\sum_{n=1}^{\infty} b_{n}$