Series Blueprinting

This advanced sequence introduces powerful tools for analyzing series with factorials and powers, leading to the concept of power series. Students master the Ratio and Root tests, explore absolute versus conditional convergence, and conclude by connecting series to functions through Taylor polynomials.

The Ratio Test

Students learn the most versatile test for series involving factorials and exponentials. They practice setting up the ratio of adjacent terms and taking the limit, while exploring why the test is inconclusive when the limit equals 1.

The Root Test

Focused on series with n-th powers, students learn to apply the Root Test. They compare the utility of Root vs. Ratio tests for different algebraic structures and practice handling limits with variable exponents.

Absolute vs Conditional Convergence

Students distinguish between series that converge 'strongly' (absolute) vs. those that barely converge (conditional). They explore the Riemann Rearrangement Theorem and how rearranging terms in the alternating harmonic series can change its sum.

Constructing Taylor Polynomials

The sequence concludes with an introduction to Taylor polynomials as approximations of functions. Students calculate coefficients using derivatives to match the slope and concavity of target functions.

Introduction to Power Series

Students transition from series of numbers to series of functions involving variables (x). They learn to find the Interval and Radius of Convergence using the Ratio Test, effectively turning series into functions.