91Ó°ÊÓ

If the power series \(f(x)=\sum c_{k} x^{k}\) has an interval of convergence of \(|x|

Use the remainder to find a bound on the error in the following approximations on the given interval. Error bounds are not unique. $$\sin x \approx x-x^{3} / 6 \text { on }[-\pi / 4, \pi / 4]$$

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

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

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. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. $$f(x)=\frac{e^{x}+e^{-x}}{2}$$

Use the remainder to find a bound on the error in the following approximations on the given interval. Error bounds are not unique. $$e^{x} \approx 1+x+x^{2} / 2 \text { on }\left[-\frac{1}{2}, \frac{1}{2}\right]$$

Find the function represented by the following series and find the interval of convergence of the series. (Not all these series are power series.) $$\sum_{k=0}^{\infty}(\sqrt{x}-2)^{k}$$

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. You do not need to use the definition of the Taylor series coefficients. b. Determine the radius of convergence of the series. $$f(x)=\left\\{\begin{array}{ll} \frac{\sin x}{x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0 \end{array}\right.$$