It is easy to show that f(0) = a0.
Then it is easy to see that f ”(0) = 2a2 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 an 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.