91Ó°ÊÓ

Representing functions by power series Identify the functions represented by the following power series. $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{k}$$

Use the Maclaurin series $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1

Representing functions by power series Identify the functions represented by the following power series. $$\sum_{k=2}^{\infty} \frac{k(k-1) x^{k}}{3^{k}}$$

Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim R_{n}(x)=0\) for all \(x\) in the interval of convergence. $$\quad f(x)=\sin x, a=0$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The interval of convergence of the power series \(\Sigma c_{k}(x-3)^{k}\) could be (-2,8) b. The series \(\sum_{k=0}^{\infty}(-2 x)^{k}\) converges on the interval \(-\frac{1}{2}

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

Representing functions by power series Identify the functions represented by the following power series. $$\sum_{k=2}^{\infty} \frac{x^{k}}{k(k-1)}$$

Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim R_{n}(x)=0\) for all \(x\) in the interval of convergence. $$f(x)=\cos 2 x, a=0$$

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

Find the remainder in the Taylor series centered at the point a for the following functions. Then show that \(\lim R_{n}(x)=0\) for all \(x\) in the interval of convergence. $$f(x)=e^{-x}, a=0$$