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• Notes

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