power series expansion

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A series of the form a ₀+a ₁ x +a ₂ x +a ₃ x +a ₄ x+………….+a n  x ⁿ+….. is called a power series expansion of a function f (x) , Where a₀,  a₁,  a₂,  a₃,  a₄………….a n,….. are constant and x is a variable.

We determine  the coefficient a₀,  a₁,  a₂,  a₃,  a₄………….a n,……. to specify power series by finding the successive derivative of the power series and evaluating them at x = 0

that is,

f(x)=a_{0}+a_{1}x+a_{2}x+a_{3}x+a_{4}x+.......a_{n}x^{n}+.......f(0)=a_{0}

f{}'(x)=a_{1}+2a_{2}x+3a_{3}x^{2}+4a_{4}x^{3}+.......+na_{n}x^{n-1}+.......f{}'(0)=a_{1}

f{{}'}'(x)= 2a_{2}+6a_{3}x+12a_{4}x^{2}+20a_{5}x_{3}+...+n(n-1)+..f(0)'=2a_{2}

f{{{}'}'}'(x)=6a_{3}+24a_{4}x+60a_{5}x^{2}+.......f{{{}'}'}'(0)=6a_{3}

f^{4}(x)=24a_{4}+120a_{5}x+.....f^{4}(0)=24a_{4}

We assume the result of this formation

(1)      a_{0}= f(0)

(2)        a_{1}=f{}'(0)

(3)         a_{2}=\frac{f{{}'}'(0)}{2!}

(4)    a_{3}=\frac{f{{{}'}'}'(0)}{3!}

(5)    a_{4}=\frac{f{}'{{{}'}'}'(0)}{4!}

And so on we get the final form

a_{n}=\frac{f^{n}(0)}{n!}

Power series expansion is final.

Thus substituting these values in the power series , we have

f(x)=f(0)+f{}'(0)x+\frac{f{{}'}'(0)}{2!}x^{2}+\frac{f{}'{{}'}'(0)}{3!}x^{3}+...+\frac{f^{n}(0)}{n!}x^{n}+....

This Expansion of f(x) is called the MacLaurin series expansion.

And also named as MacLaurin theorem and can be stated as;