Chapter 10

How are the Taylor polynomials for a function \(f\) centered at \(a\) related to the Taylor series for the function \(f\) centered at \(a ?\)

Suppose you use a Taylor polynomial with \(n=2\) centered at 0 to approximate a function \(f\). What matching conditions are satisfied by the polynomial?

What conditions must be satisfied by a function \(f\) to have a Taylor series centered at \(a ?\)

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

How would you approximate \(e^{-0.6}\) using the Taylor series for \(e^{x} ?\)

How do you find the coefficients of the Taylor series for \(f\) centered at \(a ?\)

What tests are used to determine the radius of convergence of a power series?

The first three Taylor polynomials for \(f(x)=\sqrt{1+x}\) centered at 0 are \(p_{0}(x)=1, p_{1}(x)=1+\frac{x}{2},\) and \(p_{2}(x)=1+\frac{x}{2}-\frac{x^{2}}{8}\) Find three approximations to \(\sqrt{1.1}\)

Suggest a Taylor series and a method for approximating \(\pi.\)

How do you find the interval of convergence of a Taylor series?

Explain why a power series is tested for absolute convergence.

In general, how many terms do the Taylor polynomials \(p_{2}\) and \(p_{3}\) have in common?

If \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) and the series converges for \(|x|

Suppose you know the Maclaurin series for \(f\) and it converges for \(|x| < 1 .\) How do you find the Maclaurin series for \(f\left(x^{2}\right)\) and where does it converge?

Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain.

How is the remainder in a Taylor polynomial defined?

What condition must be met by a function \(f\) for it to have a Taylor series centered at \(a ?\)

What is the radius of convergence of the power series \(\sum c_{k}(x / 2)^{k}\) if the radius of convergence of \(\Sigma c_{k} x^{k}\) is \(R ?\)

Explain how to estimate the remainder in an approximation given by a Taylor polynomial.

In terms of the remainder, what does it mean for a Taylor series for a function \(f\) to converge to \(f ?\)

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}$$

What is the interval of convergence of the power series \(\Sigma(4 x)^{k} ?\)

Write the Maclaurin series for \(e^{2 x}\)

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{\tan ^{-1} x-x}{x^{3}}$$

How are the radii of convergence of the power series \(\sum c_{k} x^{k}\) and \(\sum(-1)^{k} c_{k} x^{k}\) related?

a. Find the linear approximating polynomial for the following functions centered at the given point a. b. Find the quadratic approximating polynomial for the following functions centered at the given point a. c. Use the polynomials obtained in parts (a) and (b) to approximate the given quantity. \(f(x)=\frac{1}{x}, a=1 ;\) approximate \(\frac{1}{1.05}\)

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=e^{-x}$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{-x-\ln (1-x)}{x^{2}}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum(2 x)^{k}$$

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\cos 2 x$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{(2 x)^{k}}{k !}$$

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\left(1+x^{2}\right)^{-1}$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{x}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{(x-1)^{k}}{k}$$

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\ln (1+x)$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{1+x-e^{x}}{4 x^{2}}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{(x-1)^{k}}{k !}$$

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=e^{2 x}$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{2 \cos 2 x-2+4 x^{2}}{2 x^{4}}$$

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=(1+2 x)^{-1}$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow \infty} x \sin \frac{1}{x}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum k !(x-10)^{k}$$

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\tan ^{-1} x$$

Evaluate the following limits using Taylor series. $$\lim _{x \rightarrow 0} \frac{3 \tan x-3 x-x^{3}}{x^{5}}$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \sin ^{k}\left(\frac{1}{k}\right) x^{k}$$

a. Find the nth-order Taylor polynomials of the given function centered at \(0,\) for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=\cos x$$

a. Find the first four nonzero terms of the Maclaurin series for the given function. b. Write the power series using summation notation. c. Determine the interval of convergence of the series. $$f(x)=\sin 3 x$$

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{2^{k}(x-3)^{k}}{k}$$

a. Find the nth-order Taylor polynomials of the given function centered at \(0,\) for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=e^{-x}$$