What is a Taylor series?

Oct 18, 2014

The Taylor series of a function is a power series, all of whose derivatives match their corresponding derivatives of the function.

Let us derive the Taylor series of a function $f \left(x\right)$, centered at $c$.

Let

$f \left(x\right) = {\sum}_{n = 0}^{\infty} {a}_{n} {\left(x - c\right)}^{n}$

$= {a}_{0} + {a}_{1} \left(x - c\right) + {a}_{2} {\left(x - c\right)}^{2} + \cdots$,

where coefficients ${a}_{1} , {a}_{2} , {a}_{3} , \ldots$ are to be determined.

By taking the derivatives,

$f ' \left(x\right) = {a}_{1} + 2 {a}_{2} \left(x - c\right) + 3 {a}_{3} {\left(x - c\right)}^{2} + \cdots$

$f ' ' \left(x\right) = 2 {a}_{2} + 3 \cdot 2 {a}_{3} \left(x - c\right) + 4 \cdot 3 {a}_{4} {\left(x - c\right)}^{2} + \cdots$

$f ' ' ' \left(x\right) = 3 \cdot 2 {a}_{3} + 4 \cdot 3 \cdot 2 {a}_{4} \left(x - c\right) + 5 \cdot 4 \cdot 3 {a}_{5} {\left(x - c\right)}^{2} + \cdots$
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By plugging in $x = c$,

f(c)=a_0=0! cdot a_0

f'(c)=a_1=1! cdot a_1

f''(c)=2a_2=2! cdot a_2

f'''(c)=3cdot2 a_3=3! cdot a_3
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f^{(n)}(c)=n! cdot a_n

By dividing by n!,

a_n={f^{(n)}(c)}/{n!}

Hence, we have the Taylor series of $f \left(x\right)$, centered at $c$

f(x)=sum_{n=0}^infty{f^{(n)}(c)}/{n!}(x-c)^n.

I hope that this was helpful.