§8.5 Finding and using Taylor series

Introduction

Using substitution and algebra to find new Taylor series expressions

Differentiating and integrating Taylor series

Let

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Note that we can write $g(x) = (1+x^2)^{-1}$. Determine $g'(x)$, $g''(x)$, and $g'''(x)$.

Find the Taylor series expansion for $g(x) = x^4 \cos(x^3)$ and determine the set of all $x$-values for which the series converges.

$g(x) = x^3 \sin(x^2)$

$h(x) = e^{-x^4}$

$p(x) = \dfrac{1}{1+5x}$

$q(x) = x^2\ln(1+x^4)$

$r(x) = \dfrac{e^{3x}-1}{3x}$

Use the familiar Taylor series for $g(x) = \frac{1}{1-x}$ to develop the Taylor series for $h(x) = \ln(1+x)$ using the Power Series Differentiation and Integration Theorem.

In this activity we determine the Taylor series expansion for $\sin(x)$ in a different way and then also find the Taylor series for $\arctan(x)$.

In this activity we determine the Taylor series expansion for $\sin(x)$ in a different way and then also find the Taylor series for $\arctan(x)$.

Use the Taylor series for $e^{x}$ to find the Taylor series for $e^{-t^2}$.

Next, evaluate the integral $\int_0^x e^{-t^2} \, dt$ by replacing $e^{-t^2}$ with its Taylor series.

Use your work in (b) to state the Taylor series for $\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt$

For what interval of $x$-values will the Taylor series for $\operatorname{erf}(x)$ converge? Why?

In probability theory, $\operatorname{erf}(x)$ is important because of its connection to the normal distribution, which is represented by a bell curve. Indeed, $\operatorname{erf}(x)$ represents the fraction of a normally distributed characteristic in a population that lies between $0$ and $x$. How can you use your result in (c) to estimate $\operatorname{erf}(0.5)$?