91Ó°ÊÓ

Find the remainder \(R_{n}\) for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of \(n\). $$f(x)=e^{-x}, a=0$$

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

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

Find the remainder \(R_{n}\) for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of \(n\). $$f(x)=\cos x, a=\frac{\pi}{2}$$

Find the remainder \(R_{n}\) for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of \(n\). $$f(x)=\sin x, a=\frac{\pi}{2}$$

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$e^{2}$$

a.Find the first four nonzero terms of the binomial series centered at 0 for the given function. b. Use the first four terms of the series to approximate the given quantify. \(f(x)=(1+x)^{-2} ;\) approximate \(\frac{1}{1.21}=\frac{1}{1.1^{2}}\).

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. $$p(x)=\frac{4 x^{12}}{1-x}$$

Find the remainder \(R_{n}\) for the nth-order Taylor polynomial centered at a for the given functions. Express the result for a general value of \(n\). $$f(x)=\frac{1}{1-x}, a=0$$

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. $$\sqrt{e}$$