Chapter 10

Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k}$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\cos 2 x+2 \sin x$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty}\left(x^{2}+1\right)^{2 k}$$

Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. $$\cos x \approx 1-x^{2} / 2 ;[-\pi / 4, \pi / 4]$$

Identify the functions represented by the following power series. $$\sum_{k=2}^{\infty} \frac{k(k-1) x^{k}}{3^{k}}$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\frac{e^{x}+e^{-x}}{2}$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty}(\sqrt{x}-2)^{k}$$

Identify the functions represented by the following power series. $$\sum_{k=2}^{\infty} \frac{x^{k}}{k(k-1)}$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{4 k}$$

Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. $$\tan x \approx x ;[-\pi / 6, \pi / 6]$$

Determine whether the following statements are true and give an explanation or counterexample. a. To evaluate \(\int_{0}^{2} \frac{d x}{1-x},\) one could expand the integrand in a Taylor series and integrate term by term. b. To approximate \(\pi / 3,\) one could substitute \(x=\sqrt{3}\) into the Taylor series for \(\tan ^{-1} x\) c. \(\sum_{k=0}^{\infty} \frac{(\ln 2)^{k}}{k !}=2\)

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\left(1+x^{2}\right)^{-2 / 3}$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty} e^{-k x}$$

Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. $$\ln (1+x) \approx x-x^{2} / 2 ;[-0.2,0.2]$$

Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). $$\lim _{x \rightarrow 0} \frac{e^{a x}-1}{x}$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\tan x$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=1}^{\infty} \frac{(x-2)^{k}}{3^{2 k}}$$

Use the remainder term to estimate the maximum error in the following approximations on the given interval. Error bounds are not unique. $$\sqrt{1+x} \approx 1+x / 2 ;[-0.1,0.1]$$

Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). $$\lim _{x \rightarrow 0} \frac{\sin a x}{\sin b x}$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\sqrt{1-x^{2}}$$

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than \(10^{-3}\) ? (The answer depends on your choice of a center.) $$e^{-0.5}$$

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty}\left(\frac{x^{2}-1}{3}\right)^{k}$$

Use Taylor series to evaluate the following limits. Express the result in terms of the parameter(s). $$\lim _{x \rightarrow 0} \frac{\sin a x-\tan a x}{b x^{3}}$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=b^{x}, \text { for } b > 0, b \neq 1$$

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than \(10^{-3}\) ? (The answer depends on your choice of a center.) $$\sin 0.2$$

Replace \(x\) by \(x-1\) in the series \(\ln (1+x)=\sum_{k=1}^{\infty} \frac{(-1)^{k+1} x^{k}}{k}\) to obtain a power series for \(\ln x\) centered at \(x=1 .\) What is the interval of convergence for the new power series?

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\frac{1}{x^{4}+2 x^{2}+1}$$

Exponential function In Section 3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty < x < \infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=e^{-x}$$

The inverse hyperbolic sine is defined in several ways; among them are $$\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}}$$ Find the first four terms of the Taylor series for \(\sinh ^{-1} x\) using these two definitions (and be sure they agree).

Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. $$f(x)=\sqrt{x} \text { with } a=36 ; \text { approximate } \sqrt{39}$$

Exponential function In Section 3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty < x < \infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=e^{2 x}$$

Exponential function In Section 3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty < x < \infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=e^{-3 x}$$

Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. $$f(x)=1 / \sqrt{x} \text { with } a=4 ; \text { approximate } 1 / \sqrt{3}$$

Exponential function In Section 3, we show that the power series for the exponential function centered at 0 is $$e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !}, \quad \text { for }-\infty < x < \infty$$ Use the methods of this section to find the power series for the following functions. Give the interval of convergence for the resulting series. $$f(x)=x^{2} e^{x}$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The Taylor polynomials for \(f(x)=e^{-2 x}\) centered at 0 consist of even powers only. b. For \(f(x)=x^{5}-1,\) the Taylor polynomial of order 10 centered at \(x=0\) is \(f\) itself. c. The \(n\) th-order Taylor polynomial for \(f(x)=\sqrt{1+x^{2}}\) centered at 0 consists of even powers of \(x\) only.

Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. $$f(x)=1 / \sqrt{x} \text { with } a=4 ; \text { approximate } 1 / \sqrt{3}$$

Prove that if \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) converges on the interval \(I,\) then the power series for \(x^{m} f(x)\) also converges on \(I\) for positive integers \(m\)

Recall that the Taylor series for \(f(x)=1 /(1-x)\) about 0 is the geometric series \(\sum_{k=0}^{\infty} x^{k} .\) Show that this series can also be found as a case of the binomial series.

The expected (average) number of tosses of a fair coin required to obtain the first head is \(\sum_{k=1}^{\infty} k\left(\frac{1}{2}\right)^{k} .\) Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)

Show that the coefficients in the Taylor series (binomial series) for \(f(x)=\sqrt{1+4 x}\) about 0 are integers.

Let $$f(x)=\sum_{k=0}^{\infty} c_{k} x^{k} \quad \text { and } \quad g(x)=\sum_{k=0}^{\infty} d_{k} x^{k}$$ a. Multiply the power series together as if they were polynomials, collecting all terms that are multiples of \(1, x,\) and \(x^{2} .\) Write the first three terms of the product \(f(x) g(x)\) b. Find a general expression for the coefficient of \(x^{n}\) in the product series, for \(n=0,1,2, \ldots\)

Teams \(A\) and \(B\) go into sudden death overtime after playing to a tie. The teams alternate possession of the ball and the first team to score wins. Each team has a \(\frac{1}{6}\) chance of scoring when it has the ball, with Team \(\mathrm{A}\) having the ball first. a. The probability that Team A ultimately wins is \(\sum_{k=0}^{\infty} \frac{1}{6}\left(\frac{5}{6}\right)^{2 k}\) Evaluate this series. b. The expected number of rounds (possessions by either team) required for the overtime to end is \(\frac{1}{6} \sum_{k=1}^{\infty} k\left(\frac{5}{6}\right)^{k-1} .\) Evaluate this series.

Given the power series $$\frac{1}{\sqrt{1-x^{2}}}=1+\frac{1}{2} x^{2}+\frac{1 \cdot 3}{2 \cdot 4} x^{4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{6}+\cdots$$ for \(-1< x <1,\) find the power series for \(f(x)=\sin ^{-1} x\) centered at \(0 .\)

The period of a pendulum is given by $$T=4 \sqrt{\frac{\ell}{g}} \int_{0}^{\pi / 2} \frac{d \theta}{\sqrt{1-k^{2} \sin ^{2} \theta}}=4 \sqrt{\frac{\ell}{g}} F(k)$$ where \(\ell\) is the length of the pendulum, \(g \approx 9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, \(k=\sin \left(\theta_{0} / 2\right),\) and \(\theta_{0}\) is the initial angular displacement of the pendulum (in radians). The integral in this formula \(F(k)\) is called an elliptic integral and it cannot be evaluated analytically. a. Approximate \(F(0.1)\) by expanding the integrand in a Taylor (binomial) series and integrating term by term. b. How many terms of the Taylor series do you suggest using to obtain an approximation to \(F(0.1)\) with an error less than \(10^{-3} ?\) c. Would you expect to use fewer or more terms (than in part (b)) to approximate \(F(0.2)\) to the same accuracy? Explain.

By comparing the first four terms, show that the Maclaurin series for \(\sin ^{2} x\) can be found (a) by squaring the Maclaurin series for \(\sin x,\) (b) by using the identity \(\sin ^{2} x=(1-\cos 2 x) / 2,\) or \((\mathrm{c})\) by computing the coefficients using the definition.

Consider the following function and its power series: $$ f(x)=\frac{1}{(1-x)^{2}}=\sum_{k=1}^{\infty} k x^{k-1}, \quad \text { for }-1

Consider the following common approximations when \(x\) is near zero. a. Estimate \(f(0.1)\) and give the maximum error in the approximation. b. Estimate \(f(0.2)\) and give the maximum error in the approximation. $$f(x)=\tan x \approx x$$

The function \(\operatorname{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) is called the sine integral function. a. Expand the integrand in a Taylor series about 0 . b. Integrate the series to find a Taylor series for Si. c. Approximate Si(0.5) and Si(1). Use enough terms of the series so the error in the approximation does not exceed \(10^{-3}\).

By comparing the first four terms, show that the Maclaurin series for \(\cos ^{2} x\) can be found (a) by squaring the Maclaurin series for \(\cos x,\) (b) by using the identity \(\cos ^{2} x=(1+\cos 2 x) / 2,\) or \((\mathrm{c})\) by computing the coefficients using the definition.

The theory of optics gives rise to the two Fresnel integrals $$S(x)=\int_{0}^{x} \sin t^{2} d t \text { and } C(x)=\int_{0}^{x} \cos t^{2} d t$$ a. Compute \(S^{\prime}(x)\) and \(C^{\prime}(x)\) b. Expand \(\sin t^{2}\) and \(\cos t^{2}\) in a Maclaurin series and then integrate to find the first four nonzero terms of the Maclaurin series for \(S\) and \(C\) c. Use the polynomials in part (b) to approximate \(S(0.05)\) and \(C(-0.25)\) d. How many terms of the Maclaurin series are required to approximate \(S(0.05)\) with an error no greater than \(10^{-4} ?\) e. How many terms of the Maclaurin series are required to approximate \(C(-0.25)\) with an error no greater than \(10^{-6} ?\)