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

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