Chapter 10

Use the geometric series $$f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}, \quad \text { for }|x|<1$$ to find the power series representation for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f\left(x^{3}\right)=\frac{1}{1-x^{3}}$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\cos x, a=\pi / 6$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)-y(t)=0, y(0)=2$$

Use the geometric series $$f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}, \quad \text { for }|x|<1$$ to find the power series representation for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$p(x)=\frac{4 x^{12}}{1-x}$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\sqrt{x}, a=9$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)+4 y(t)=8, y(0)=0$$

Use the geometric series $$f(x)=\frac{1}{1-x}=\sum_{k=0}^{\infty} x^{k}, \quad \text { for }|x|<1$$ to find the power series representation for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f(-4 x)=\frac{1}{1+4 x}$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\sqrt[3]{x}, a=8$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)-3 y(t)=10, y(0)=2$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f(3 x)=\ln (1-3 x)$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\ln x, a=e$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)=6 y(t)+9, y(0)=2$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$g(x)=x^{3} \ln (1-x)$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\sqrt[4]{x}, a=16$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$h(x)=x \ln (1-x)$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\tan ^{-1} x+x^{2}+1, a=1$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f\left(x^{3}\right)=\ln \left(1-x^{3}\right)$$

a. Find the nth-order Taylor polynomials for the given function centered at the given point a, for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=e^{x}, a=\ln 2$$

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}\). $$\int_{-0.35}^{0.35} \cos 2 x^{2} d x$$

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction. b. Use the first four terms of the series to approximate the given quantity. $$f(x)=(1+x)^{-2} ; \text {approximate } 1 / 1.21=1 / 1.1^{2}$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$p(x)=2 x^{6} \ln (1-x)$$

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction. b. Use the first four terms of the series to approximate the given quantity. $$f(x)=\sqrt{1+x} ; \text { approximate } \sqrt{1.06}$$

Use the power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of convergence of the new series. $$f(-4 x)=\ln (1+4 x)$$

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. $$\cos (-0.2)$$

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction. b. Use the first four terms of the series to approximate the given quantity. $$f(x)=\sqrt[4]{1+x} ; \text { approximate } \sqrt[4]{1.12}$$

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for \(f\) (perhaps more than once). Give the interval of convergence for the resulting series. $$g(x)=\frac{1}{(1-x)^{2}} \text { using } f(x)=\frac{1}{1-x}$$

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. $$\tan (-0.1)$$

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}\). $$\int_{0}^{0.4} \ln \left(1+x^{2}\right) d x$$

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction. b. Use the first four terms of the series to approximate the given quantity. $$f(x)=(1+x)^{-3} ; \text {approximate } 1 / 1.331=1 / 1.1^{3}$$

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for \(f\) (perhaps more than once). Give the interval of convergence for the resulting series. $$g(x)=\frac{1}{(1-x)^{3}} \text { using } f(x)=\frac{1}{1-x}$$

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. $$\ln (1.05)$$

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction. b. Use the first four terms of the series to approximate the given quantity. $$f(x)=(1+x)^{-2 / 3} ; \text { approximate } 1.18^{-2 / 3}$$

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for \(f\) (perhaps more than once). Give the interval of convergence for the resulting series. $$g(x)=\frac{1}{(1-x)^{4}} \text { using } f(x)=\frac{1}{1-x}$$

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}\). $$\int_{0}^{0.2} \frac{\ln (1+t)}{t} d t$$

a. Find the first four nonzero terms of the Taylor series centered at 0 for the given finction. b. Use the first four terms of the series to approximate the given quantity. $$f(x)=(1+x)^{2 / 3} ; \text { approximate } 1.02^{2 / 3}$$

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for \(f\) (perhaps more than once). Give the interval of convergence for the resulting series. $$g(x)=\frac{x}{\left(1+x^{2}\right)^{2}} \text { using } f(x)=\frac{1}{1+x^{2}}$$

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. $$\sqrt[4]{79}$$

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$e^{2}$$

Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series. $$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots, \text { for }-1 < x \leq 1$$ $$\sqrt{1+x^{2}}$$

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for \(f\) (perhaps more than once). Give the interval of convergence for the resulting series. $$g(x)=\ln (1-3 x) \text { using } f(x)=\frac{1}{1-3 x}$$

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$\sqrt{e}$$

Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series. $$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots, \text { for }-1 < x \leq 1$$ $$\sqrt{4+x}$$

Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for \(f\) (perhaps more than once). Give the interval of convergence for the resulting series. $$g(x)=\ln \left(1+x^{2}\right) \text { using } f(x)=\frac{x}{1+x^{2}}$$

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. $$\sqrt[3]{126}$$

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$\cos 2$$

Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series. $$\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots, \text { for }-1 < x \leq 1$$ $$\sqrt{9-9 x}$$

Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. $$f(x)=\frac{1}{1+x^{2}}$$

a. Approximate the given quantities using Taylor polynomials with \(n=3\) b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. $$\sinh (0.5)$$

Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$\sin 1$$

Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. $$f(x)=\frac{1}{1-x^{4}}$$