Lesson 11: Power Series

Unit Preview

• This lesson marks the start of Unit 3: Power Series and Taylor Series. The driving question of this unit is: Can a function be approximated by a polynomial? That is, can we say that f(x) is approximately p(x), where f is a function and p a polynomial? We know from Calculus 1—and the theory of linear approximation, specifically—that if f is differentiable then f(x) is approximately L(x), where L is the equation of the tangent line at x = a, with this approximation being most accurate for x near a. But is this the best we can do? Can we get higher-accuracy approximations? We’ll soon find out that in many cases the answer is: yes! Furthermore, we’ll discover that in some cases we’ll be able to say that f(x) = sum p_n(x), where the p_n are special polynomials. Working all that out, and exploring the implications and applications of what we’ll develop, constitutes the bulk of this unit.

Lesson Preview

• In this lesson we’ll lay the foundations for exploring our new driving question. In Module 1 we’ll return to a particular geometric series we studied in Lesson 6 and use it to motivate the topic of power series—infinite series whose terms are of the form a_n = c_n(x-a)^n, where a and the c_n are real numbers. These power series yield the infinite sums of polynomials we’ll eventually use (in Lessons 13–15) to achieve the feats described in the Unit preview above. Finally, in Module 2 we’ll discuss the convergence/divergence of these new types of series (power series).