I’m beginning to feel like there is more to know than I will ever have time for…during our study session last week I also realized that my Taylor series knowledge was pretty rusty. I wanted to write a very short post on how the Taylor series is derived to help me remember it in case I get asked in the exam.

The basic concept of a Taylor series is that you can approximate any function as a sum of length n of the function’s n-th derivatives [1]-[3]. The approximation will, theoretically, be a perfect match if n is equal to infinity [1]-[3]. The Taylor series can be used to approximate the function at a chosen point x = c [1]-[3]. The coefficient is chosen to cancel out the coefficients of the derivatives [1]-[3].

The general expression for a Taylor series is given below [1]-[3].

References:

[1] Khan Academy. “Taylor & Maclaurin polynomials intro (part 1).” https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-11/v/maclaurin-and-taylor-series-intuition Visited 09/29/2019.

[2] Khan Academy. “Taylor & Maclaurin polynomials intro (part 2).” https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-11/v/generalized-taylor-series-approximation Visited 09/29/2019.

[3] Khan Academy. “Worked example: coefficient in Taylor polynomial.” https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-11/v/fourth-degree-coefficient-for-taylor-polynomial Visited 09/29/2019.