McLaurin’s Series

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

f(x) = Power_Series

It is easy to show that  f(0) = a0.

And, since f ‘(x) = D1_PS ,  it follows that:

zero_eval2    and   f ”(x) = D2_PS

Then it is easy to see that  f ”(0) = 2a2  and

f ”'(x) = D3_PS

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:

zero_eval_n .

Solving for an gives:

McLaurin_an .

Substituting in the original series representation for f we obtain:

McLaurin_formula

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.