Chapter 10

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=3^{x}$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{3 \tan ^{-1} x-3 x+x^{3}}{x^{5}}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum\left(\frac{x}{3}\right)^{k}$$

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

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\log _{3}(x+1)$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1-(x / 2)}{4 x^{2}}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum(-1)^{k} \frac{x^{k}}{5^{k}}$$

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

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\cosh x$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{\sin x-\tan x}{3 x^{3} \cos x}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{x^{k}}{k^{k}}$$

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

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\sinh 2 x$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 1} \frac{x-1}{\ln x}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum(-1)^{k} \frac{k(x-4)^{k}}{2^{k}}$$

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

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=\sin x, a=\pi / 2$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{k^{2} x^{2 k}}{k !}$$

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

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=\cos x, a=\pi$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow \infty} x\left(e^{1 / x}-1\right)$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum k(x-1)^{k}$$

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=1 / x, a=1$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0^{+}} \frac{(1+x)^{-2}-4 e^{(-x / 2)}+3}{2 x^{2}}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{x^{2 k+1}}{3^{k-1}}$$

a. Use the given Taylor polynomial \(p_{2}\) to approximate the given quantity. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. Approximate \(\sqrt{1.05}\) using \(f(x)=\sqrt{1+x}\) and \(p_{2}(x)=1+x / 2-x^{2} / 8\)

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=1 / x, a=2$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{(1-2 x)^{-1 / 2}-e^{x}}{8 x^{2}}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum\left(-\frac{x}{10}\right)^{2 k}$$

a. Use the given Taylor polynomial \(p_{2}\) to approximate the given quantity. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. Approximate \(\sqrt[3]{1.1}\) using \(f(x)=\sqrt[3]{1+x}\) and \(p_{2}(x)=1+x / 3-x^{2} / 9\)

a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. $$f(x)=e^{x}$$

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=\ln x, a=3$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{(x-1)^{k} k^{k}}{(k+1)^{k}}$$

a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. $$f(x)=\cos x$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{(-2)^{k}(x+3)^{k}}{3^{k+1}}$$

a. Use the given Taylor polynomial \(p_{2}\) to approximate the given quantity. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. Approximate ln 1.06 using \(f(x)=\ln (1+x)\) and \(p_{2}(x)=x-x^{2} / 2\)

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=2^{x}, a=1$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{k^{20} x^{k}}{(2 k+1) !}$$

a. Use the given Taylor polynomial \(p_{2}\) to approximate the given quantity. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. Approximate \(e^{-0.15}\) using \(f(x)=e^{-x}\) and \(p_{2}(x)=1-x+x^{2} / 2\).

a. Find the first four nonzero terms of the Taylor series for the given function centered at \(a\) b. Write the power series using summation notation. $$f(x)=10^{x}, a=2$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum(-1)^{k} \frac{x^{3 k}}{27^{k}}$$

a. Use the given Taylor polynomial \(p_{2}\) to approximate the given quantity. b. Compute the absolute error in the approximation assuming the exact value is given by a calculator. Approximate \(\frac{1}{1.12^{3}}\) using \(f(x)=\frac{1}{(1+x)^{3}}\) and \(p_{2}(x)=1-3 x+6 x^{2}\)

a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. $$f(x)=e^{-2 x}$$

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. $$f(3 x)=\frac{1}{1-3 x}$$

a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. $$f(x)=\sqrt{1+x}$$

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. $$g(x)=\frac{x^{3}}{1-x}$$

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. $$f(x)=8 \sqrt{x}, a=1$$

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. $$h(x)=\frac{2 x^{3}}{1-x}$$

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. $$f(x)=\sin x, a=\pi / 4$$

a. Differentiate the Taylor series about 0 for the following functions. b. Identify the function represented by the differentiated series. c. Give the interval of convergence of the power series for the derivative. $$f(x)=-\ln (1-x)$$