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,

We assume the result of this formation

(1)

(2)

(3)

(4)

(5)

And so on we get the final form

Power series expansion is final.

Thus substituting these values in the power series , we have

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

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