91Ó°ÊÓ

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. $$f(x)=2^{x}, a=1$$

Determine the radius and interval of convergence of the following power series. $$\sum_{k=0}^{\infty} \frac{(-2)^{k}(x+3)^{k}}{3^{k+1}}$$

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. $$f(x)=x \ln x-x+1 ; a=1$$

Approximations with Taylor polynomials 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. $$\cos (-0.2)$$

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

Determine the radius and interval of convergence of the following power series. $$\sum_{k=0}^{\infty} \frac{k^{20} x^{k}}{(2 k+1) !}$$

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

Determine the radius and interval of convergence of the following power series. $$\sum_{k=0}^{\infty}(-1)^{k} \frac{x^{3 k}}{27^{k}}$$

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

Use the Taylor series in Table 11 to find the first four nonzero terms of the Taylor series for the following functions centered at \(0 .\) $$\frac{1}{1-2 x}$$