Let be a power series representation of f(x) within the interval of convergence |x| < r. That is,

It is easy to show that f(0) = a_{0}.

And, since f ‘(x) = , it follows that:

Then it is easy to see that f ”(0) = 2a_{2} and

But 3⋅2 = 3(2)(1) = 3! and 4⋅3⋅2 = 4(3)(2)(1) = 4!

This pattern indicates that if we differentiate the power series for f(x) **n** times, and set x = 0, we will have:

Solving for **a _{n}** gives:

Substituting in the original series representation for f we obtain:

This is known as McLaurin’s Series, to honor its discoverer, the Scottish mathematician Colin Mclaurin.

By computing the sum of a finite number of terms of McLaurin’s series for f we obtain an approximation to the actual value of f(x). When x is small such approximations may be very accurate.