§8.4 Taylor series

Introduction

Taylor series and the Ratio Test

Taylor series of several important functions

Summary

Let

.

Explain why $T(x)$ is a geometric series and identify the values of $a$ and $r$.

As described in the statement of the Ratio Test, let $r_n(x)$ be the ratio of the $(n+1)^{\text{st}}$ term of $T(x)$ to the $n^{\text{th}}$ term of $T(x)$. Find the simplest formula that you can for $r_n(x)$.

Let $r(x) = \lim_{n \to \infty} r_n(x)$. Evaluate this limit to find the simplest formula you can for $r(x)$.

For what values of $x$ is $|r(x)| \lt 1$? What is the open interval of convergence for $T(x)$?

Let $T_{10}(x)$ be the sum of the first $10$ terms of $T(x)$, and let $f(x) = \ln(2) - \ln(3-x)$. Plot $f(x)$ and $T_{10}(x)$ on the same coordinate axes in a window centered around $x=1$. What do you notice? What does this suggest about the series $T(x)$?

Explain why the Taylor series centered at $a = 0$ for $f(x) = \sin(x)$ is

and find the interval of $x$-values for which this Taylor series converges. Investigate whether or not the Taylor series converges to $f(x) = \sin(x)$.

For $f(x) = e^x$, explain why $f^{(k)}(0) = 1$ for every natural number $k$.

State the Taylor series, $T_f(x)$ centered at $a = 0$ for $f(x) = e^x$. Write $T_f(x)$ in both sigma notation and as an expanded sum.

Let $r_n(x)$ be the ratio of the $(n+1)^{\text{st}}$ term to the $n^{\text{th}}$ term of $T_f(x)$. Find the simplest expression you can for $r_n(x)$.

Let $r(x) = \lim_{n \to \infty} r_n(x)$. Evaluate this limit, and then apply the Ratio Test to say what you can conclude about the $x$-values for which $T_f(x)$ converges.

Use a computational device to graph $f(x) = e^x$, $T_{10}(x)$, and $T_{20}(x)$ on the same axes. What do you observe?

The examples we have considered so far in this section have all been for Taylor polynomials and series centered at 0, but Taylor polynomials and series can be centered at any value of $a$.

• Let $f(x) = \cos(x)$. Find the Taylor polynomials up through order four of $f$ centered at $a = \frac{\pi}{2}$. Then find the Taylor series for $f(x)$ centered at $a = \frac{\pi}{2}$. Why is the result not surprising? - Let $f(x) = \frac{1}{1+x} = (1+x)^{-1}$. Find the Taylor polynomials up through order four of $f$ centered at $a = 1$. Then find the Taylor series for $f(x)$ centered at $a = 1$.

As we will see in more detail in the next section, we can use known Taylor series to obtain other Taylor series, and we preview that idea in this exercise.

• Calculate the first four derivatives of $\sin(x^2)$ and hence find the fourth order Taylor polynomial for $\sin(x^2)$ centered at $a=0$. - Part (a) demonstrates the direct approach to finding Taylor polynomials and series. Next we utilize a known Taylor series to make the process simpler. Recall that the Taylor series centered at 0 for $f(x) = \sin(x)$ is

. - Substitute $x^2$ for $x$ in the Taylor series $T(x)$ in Equation. Write out the first several terms and compare to your work in part (a). Explain why the substitution in this problem should result in the Taylor series for $\sin(x^2)$ centered at 0. - For what interval of $x$-values should we expect the Taylor series for $\sin(x^2)$ to converge?