91Ó°ÊÓ

Determine the radius and interval of convergence of the following power series. $$\sum_{k=2}^{\infty} \frac{(x+3)^{k}}{k \ln ^{2} k}$$

Power series for derivatives a. Differentiate the Taylor series centered at 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. $$f(x)=e^{-2 x}$$

Approximations with Taylor polynomials 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. Use the definition of a Taylor series to 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. $$f(x)=\frac{1}{x}, a=1$$

Determine the radius and interval of convergence of the following power series. $$\sum_{k=0}^{\infty} \frac{k^{2} x^{2 k}}{k !}$$

Determine the radius and interval of convergence of the following power series. $$\sum_{k=0}^{\infty} k(x-1)^{k}$$

Approximations with Taylor polynomials 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 \(1 / \sqrt{1.08}\) using \(f(x)=1 / \sqrt{1+x}\) and \(p_{2}(x)=1-x / 2+3 x^{2} / 8\).

a. Use the definition of a Taylor series to 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. $$f(x)=\frac{1}{x}, a=2$$

Power series for derivatives a. Differentiate the Taylor series centered at 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. $$f(x)=(1-x)^{-1}$$

Determine the radius and interval of convergence of the following power series. $$\sum_{k=1}^{\infty} \frac{x^{2 k+1}}{3^{k-1}}$$