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\begin{aligned} f(x) &=\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n !}(x-a)^{n} \\ &=f(a)+\frac{f^{\prime}(a)}{1 !}(x-a)+\frac{f^{n}(a)}{2 !}(x-a)^{2}+\frac{f^{\prime \prime \prime}(a)}{3 !}(x-a)^{3}+\cdots \end{aligned}

The series in is called the Taylor series of the function $f$ at $a$ (or about

$a$ or centered at $a$)

$f(x)=\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n !} x^{n}=f(0)+\frac{f^{\prime}(0)}{1 !} x+\frac{f^{\prime \prime}(0)}{2 !} x^{2}+\cdots$

This case arises frequently enough that it is given the special name Maclaurin series.

نظرية هامة جدا:

The Binomial Series If $k$ is any real number and $|x|<1,$ then

$(1+x)^{k}=\sum_{n=0}^{\infty} \left( \begin{array}{c}{k} \\ {n}\end{array}\right) x^{n}=1+k x+\frac{k(k-1)}{2 !} x^{2}+\frac{k(k-1)(k-2)}{3 !} x^{3}+\cdots$