Representation of Functions by Taylor Series and Maclaurin Series

Date Range: November 28 – December 4

Hours Spent:
12 hours  

 

• • 6 hours on  lectures and notes 1  : Professor Leonard – Calculus 2 Lecture 9.8: Representation of Functions by Taylor Series and Maclauren Series (Calculus 2 Lecture 9.8: Representation of Functions by Taylor Series and Maclaurin Series)

• • 6 hours on exercises (Thomas 13th ed, 10.8 questions 1 – 28)

Summary:

In the last section I learned that I can express power series as functions of infinite polynomials around a chosen center.  Not all functions match a simple geometric or known power series, but if a function is suitable enough, i.e. it’s infinitely differentiable at its center and an interval for the Taylor Series to equal the function, we can construct a power series for it using its derivatives.  

Building upon this idea, we arrive at Taylor Series – representations of functions as infinite sums of derivative-based terms centered around a chosen value a.  As we add on greater nth derivatives, we can generate increasingly accurate polynomial approximations of the function.  Maclaurin Series are types of Taylor Series with centers of a = 0.

We build Taylor Series as follows:

 

1. 1. Choose a center a

1. 1. Computer n derivatives of our function, and then the corresponding n derivatives for f(a)

1. 1. Plug in those values into the Taylor formula: or the Maclaurin formula if a = 0

1. 1. Evaluate what comes out

For example, a simple but fun one is where we get a beautiful Taylor Series that looks like this:  .Or we can mix in some trig functions to spice things up (you know I love to mix in trig!), like in this problem:

At first, it looks messy as the second, third, fourth, etc, derivatives start piling on, e.g.

but with a little understanding of how this trig functions behave (I’d hope I’d have some understanding by now), I can quickly find the Taylor Series.

Personal Observations:

 

• • Despite this lecture being the longest of Professor Leonard’s Calculus 2 series, I found the problems very easy to crank out – derivatives and basic algebra is all

• • I can see that Taylor Series could prove very useful in putting complex phenomena into more concrete parameters of a polynomial – I suspect Taylor Series come in very useful in Physics and Engineering, as well as my own field of interest – Economics

Reflections:
I feel pretty good with my progress, as the Taylor Series, which is a very important concept in Calc 2, was really straightforward for me.  This course has been a long slog (more than 4 months now) – I’ve come a long way.  Although I’m not quite ready to pat myself on the back, the end of this course is right around the corner.  A good pat is a coming.