is differentiable (and therefore continuous) on the interval \((a-R, a+R)\) and

(i) \(f^{\prime}(x)=c_{1}+2 c_{2}(x-a)+3 c_{3}(x-a)^{2}+\cdots=\sum_{n=1}^{\infty} n c_{n}(x-a)^{n-1}\)

(ii) \(\int f(x) d x \ and \ =C+c_{0}(x-a)+c_{1} \frac{(x-a)^{2}}{2}+c_{2} \frac{(x-a)^{3}}{3}+\cdots \ \\ and \ =C+\sum_{n=0}^{\infty} c_{n} \frac{(x-a)^{n+1}}{n+1} \)

The radii of convergence of the power series in Equations ( i) and (ii) are both R

الـ The sum of a power series هو الدالة

\(f(x)=\sum_{n=0}^{\infty} c_{n}(x-a)^{n}\)

والـdomain هو الـ interval of convergence للـ series

Theorem If the power series \(\Sigma c_{n}(x-a)^{n}\) has radius of convergence

\(R\) then the function \(f\) defined by

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

is differentiable (and therefore continuous) on the interval \((a-R, a+R)\) and

(i) \(f^{\prime}(x)=c_{1}+2 c_{2}(x-a)+3 c_{3}(x-a)^{2}+\cdots=\sum_{n=1}^{\infty} n c_{n}(x-a)^{n-1}\)

(ii) \(\int f(x) d x \ and \ =C+c_{0}(x-a)+c_{1} \frac{(x-a)^{2}}{2}+c_{2} \frac{(x-a)^{3}}{3}+\cdots \ \\ and \ =C+\sum_{n=0}^{\infty} c_{n} \frac{(x-a)^{n+1}}{n+1} \)

The radii of convergence of the power series in Equations ( i) and (ii) are both R

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