In this chapter, we're first going to see how a function can be approximated by a polynomial. Once we've gotten the approximation method down, we'll extend it to infinite series. As it turns out the function, whatever it is, will be EXACTLY the same as the series, which is pretty remarkable.
Why do we care? Transcendental functions like sin(x), cos(x), e^x, and the like are not easy to evaluate and are often intractable little beasts when it comes to integration, and integration turns out to be important in all kinds of applications (those we've seen and many, more more types). Engineering students, you'll see series again when you study advanced engineering mathematics.
Polynomial functions, on the other hand, are very easy to work with, so besides just the "gee whiz" factor, the need for efficient ways to deal with the transcendental functions led to the push (a long time ago, admittedly) for the so-called "power series".
Here is a brilliant introduction to the topics we'll be covering in this Chapter:
Video: Taylor Series/Essence of Calculus
In Section 10.1 [11.1], you'll learn how to find MacLaurin and Taylor Polynomials.
Here's an animation that shows how a polynomial starts to look like sin(x) as terms are added to the polynomial (click "movie" or "start" at the bottom): Animation
So, if sin(x) is approximated by x - x^3/3! + x^5/5! - ... (see the animation), where did all those factorials come from? The video above may have given you the idea already, but here is the Khan Academy version with follow-up examples:
Video: Taylor and MacLaurin Polynomials Intro (Part 1)
Taylor and MacLaurin Polynomials Intro (Part 2)
Helpful Practice
Note 1: The general name for these polynomials is "Taylor Polynomials" (and later, Taylor Series). "Maclaurin" polynomials and series are just a special case where the expansion is about x = 0.
Note 2: You can check your answer for a polynomial on Wolfram by filling in "taylor series [fill in function] at x = [fill in value]". Then truncate the series at the appropriate term.