Taylor Polynomials II, Part 6

Taylor Polynomials II

Part 6: Summary

Geometric polynomials with each term x times the preceding one are also Taylor polynomials for some function of x. What function? What is the interval of convergence for this sequence of Taylor polynomials? How can you be sure of your answer to the preceding question?

The Taylor polynomials for the function f(t) = 1/(1 + t) are also geometric. What is the common ratio of each term to the preceding one? What is the interval of convergence for this sequence of Taylor polynomials? How can you be sure of your answer to the preceding question?

The Taylor polynomials for the function g(x) = 1/(1 + x²) are also geometric. What is the common ratio of each term to the preceding one? What is the interval of convergence for this sequence of Taylor polynomials? How can you be sure of your answer to the preceding question?

Are the Taylor polynomials for ln(1 + x) geometric? Why or why not?

We did not have absolutely conclusive evidence for the interval of convergence of the Taylor polynomials for ln(1 + x). Describe the evidence that supports your conclusion about the interval of convergence.

Are the Taylor polynomials for arctan x geometric? Why or why not?

We did not have absolutely conclusive evidence for the interval of convergence of the Taylor polynomials for arctan x. Describe the evidence that supports your conclusion about the interval of convergence.

In Part 2 of Taylor Polynomials I, we stated the following definitions for any function f(x) that can be differentiated at least n times: The numbers

are called the Taylor coefficients of f, and the polynomial

is called the n-th degree Taylor polynomial of f. In this module we found Taylor coefficients without using this definition. Why do suppose we did not use it? Your answer should make some distinction between the functions studied in each of the two modules. [Hint: The distinction has nothing to do with intervals of convergence.]