Calculus BPower Series

الـ power series  تكون على الشكل

\(\sum_{n=0}^{\infty} c_{n} x^{n}=c_{0}+c_{1} x+c_{2} x^{2}+c_{3} x^{3}+\cdots\)

حيث  x متغير و حدود الـ  c_n  ثوابت وتُسمى بـ the coefficients of the series

الـ power series  من الممكن ان تكون converge  لبعض قيم اكس و diverge  لبعض القيم الاخرى

الـ sum   للـ series  عبارة عن دالة

\(f(x)=c_{0}+c_{1} x+c_{2} x^{2}+\cdots+c_{n} x^{n}+\cdots\)

whose domain is the set of all \(x\) for which the series converges.

Theorem For a given power series \(\sum_{n=0}^{\infty} c_{n}(x-a)^{n},\) there are only three

possibilities:

(i) The series converges only when \(x=a\) .

(ii) The series converges for all \(x\) .

(iii) The series converges for all \(x\) .

(iii) There is a positive number \(R\) such that the series converges if \(|x-a|<R\)

and diverges if \(|x-a|>R\) .