91Ó°ÊÓ

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. $$e^{0.12}$$

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 power series representation $$f(x)=\ln (1-x)=-\sum_{k=1}^{\infty} \frac{x^{k}}{k}, \quad \text { for }-1 \leq x<1$$ to find the power series for the following functions (centered at 0 ). Give the interval of comvergence of the new series. $$p(x)=2 x^{6} \ln (1-x)$$

Evaluating an infinite series Write the Taylor series for \(f(x)=\ln (1+x)\) centered at 0 and find its interval of convergence. Assume the Taylor series converges to \(f\) on the interval of convergence. Evaluate \(f(1)\) to find the value of \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k}\) (the alternating harmonic series).

Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique. $$\sqrt{1+x}=1+\frac{x}{2} \text {on } [-0.1,0.1]$$

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than \(10^{-3} ?\) (The answer depends on your choice of a center.) $$e^{-0.5}$$

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than \(10^{-3} ?\) (The answer depends on your choice of a center.) $$\cos (-0.25)$$

Limits with a parameter Use Taylor series to evaluate the follow. ing limits. Express the result in terms of the nonzero real parameter(s). $$\lim _{x \rightarrow 0} \frac{e^{a x}-1}{x}$$

Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.) $$\sum_{k=1}^{\infty} \frac{x^{2 k}}{4^{k}}$$

Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.) $$\sum_{k=0}^{\infty}\left(\frac{x^{2}-1}{3}\right)^{k}$$