Nonconvergence to f Consider the function a. Use the

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

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

Nonconvergence to f Consider the function \(f(x)=\left\{\begin{array}{ll} e^{-1 / x^{2}} & \text { if } x \neq 0 \\ 0 & \text { if } x=0 \end{array}\right.\) a. Use the definition of the derivative to show that \(f^{\prime}(0)=0\). b. Assume the fact that \(f^{(k)}(0)=0\), for k = 1, 2, 3, .... (You can write a proof using the definition of the derivative.) Write the Taylor series for f centered at 0. c. Explain why the Taylor series for f does not converge to f for \(x \neq 0\).

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

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

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?

Maclaurin series 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 2x

For what values of p does the Taylor series for \(f(x)=(1+x)^{p}\) centered at 0 terminate?

Maclaurin series 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. \(f(x)=e^{-x}\)

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

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

Maclaurin series 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. \(f(x)=\left(1+x^{2}\right)^{-1}\)

Maclaurin series 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) = In (1 + x)

Maclaurin series 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. \(f(x)=e^{2 x}\)

Maclaurin series 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. \(f(x)=\tan ^{-1} x\)

Maclaurin series 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 3x

Maclaurin series 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. \(f(x)=(1+2 x)^{-1}\)

Taylor series centered at \(a \neq 0\) a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. \(f(x)=\sin x, a=\pi / 2\)

Taylor series centered at \(a \neq 0\) a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. \(f(x)=\cos x, a=\pi\)

Taylor series centered at \(a \neq 0\) a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. f(x) = 1/x, a = 1

Taylor series centered at \(a \neq 0\) a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. f(x) = 1/x, a = 2

Taylor series centered at \(a \neq 0\) a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. \(f(x)=e^{x}, a=\ln 2\)

Manipulating Taylor series Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. \(\ln \left(1+x^{2}\right)\)

Taylor series centered at \(a \neq 0\) a. Find the first four nonzero terms of the Taylor series for the given function centered at a. b. Write the power series using summation notation. f(x) = In x, a = 3

Manipulating Taylor series Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. \(\sin x^{2}\)

Manipulating Taylor series Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. \(\frac{e^{x}-1}{x}\)

Manipulating Taylor series Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. \(\cos \sqrt{x}\)

Manipulating Taylor series Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. \(x \tan ^{-1} x^{2}\)

Manipulating Taylor series Use the Taylor series in Table 9.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0. \(\left(1+x^{4}\right)^{-1}\)

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

Binomial series a. Find the first four nonzero terms of the Taylor series centered at 0 for the given function. b. Use the first four terms of the series to approximate the given quantity. \(f(x)=\sqrt{1+x}\); approximate \(\sqrt{1.06}\).

Binomial series a. Find the first four nonzero terms of the Taylor series centered at 0 for the given function. b. Use the first four terms of the series to approximate the given quantity. \(f(x)=\sqrt[4]{1+x}\); approximate \(\sqrt[4]{1.12}\).

Binomial series a. Find the first four nonzero terms of the Taylor series centered at 0 for the given function. b. Use the first four terms of the series to approximate the given quantity. \(f(x)=(1+x)^{-2 / 3}\); approximate \(1.18^{-2 / 3}\).

Binomial series a. Find the first four nonzero terms of the Taylor series centered at 0 for the given function. b. Use the first four terms of the series to approximate the given quantity. \(f(x)=(1+x)^{-3}\); approximate \(1 / 1.331=1 / 1.1^{3}\).

Binomial series a. Find the first four nonzero terms of the Taylor series centered at 0 for the given function. b. Use the first four terms of the series to approximate the given quantity. \(f(x)=(1+x)^{2 / 3}\); approximate \(1.02^{2 / 3}\).

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series \(\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots\), for \(-1<x \leq 1\). \(\sqrt{1+x^{2}}\)

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series \(\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots\), for \(-1<x \leq 1\). \(\sqrt{4+x}\)

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series \(\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots\), for \(-1<x \leq 1\). \(\sqrt{9-9 x}\)

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series \(\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots\), for \(-1<x \leq 1\). \(\sqrt{1-4 x}\)

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series \(\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots\), for \(-1<x \leq 1\). \(\sqrt{4-16 x^{2}}\)

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Give the interval of convergence for the new series. Use the Taylor series \(\sqrt{1+x}=1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\cdots\), for \(-1<x \leq 1\). \(\sqrt{a^{2}+x^{2}}, a>0\)

Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series \((1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots\), for -1 < x < 1. \((1+4 x)^{-2}\)

Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series \((1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots\), for -1 < x < 1. \(\frac{1}{(1-4 x)^{2}}\)

Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series \((1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots\), for -1 < x < 1. \(\frac{1}{\left(4+x^{2}\right)^{2}}\)

Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series \((1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots\), for -1 < x < 1. \(\left(x^{2}-4 x+5\right)^{-2}\)

Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series \((1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots\), for -1 < x < 1. \(\frac{1}{(3+4 x)^{2}}\)

Working with binomial series Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series \((1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots\), for -1 < x < 1. \(\frac{1}{\left(1+4 x^{2}\right)^{2}}\)

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

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

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

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

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The function \(f(x)=\sqrt{x}\) has a Taylor series centered at 0. b. The function f(x) = csc x has a Taylor series centered at \(\pi / 2\). c. If f has a Taylor series that converges only on (-2,2), then \(f\left(x^{2}\right)\) has a Taylor series that also converges only on (-2,2). d. If p is the Taylor series for f centered at 0, then p(x - 1) is the Taylor series for f centered at 1. e. The Taylor series for an even function about 0 has only even powers of x.

Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. f(x) = cos (2x) + 2 sin x

Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. \(f(x)=\frac{e^{x}+e^{-x}}{2}\)

Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. \(f(x)=\sqrt{1-x^{2}}\)

Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. \(f(x)=\left(1+x^{2}\right)^{-2 / 3}\)

Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. \(f(x)=b^{x} \quad \text { for } b>0\)

Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. f(x) = tan x

Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. \(f(x)=\frac{1}{x^{4}+2 x^{2}+1}\)

Any method a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. f(x) = sec x

Alternative approach Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. \(f(x)=\sqrt{x}\) with a = 36; approximate \(\sqrt{39}\).

Alternative approach Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. \(f(x)=\sqrt[4]{x}\) with a = 16; approximate \(\sqrt[4]{13}\).

Alternative approach Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. \(f(x)=\sqrt[3]{x}\) with a = 64; approximate \(\sqrt[3]{60}\).

Alternative approach Compute the coefficients for the Taylor series for the following functions about the given point a and then use the first four terms of the series to approximate the given number. \(f(x)=1 / \sqrt{x}\) with a = 4; approximate \(1 / \sqrt{3}\).

Geometric/binomial series Recall that the Taylor series for f(x) = 1/(1 - x) about 0 is the geometric series \(\sum_{k=0}^{\infty} x^{k}\). Show that this series can also be found as a case of the binomial series.

Integer coefficients Show that the coefficients in the Taylor series (binomial series) for \(f(x)=\sqrt{1+4 x}\) about 0 are integers.

Choosing a good center Suppose you want to approximate \(\sqrt{72}\) using four terms of a Taylor series. Compare the accuracy of the approximations obtained using the Taylor series for \(\sqrt{x}\) centered at 64 and 81.

Alternative means By comparing the first four terms, show that the Maclaurin series for \(\sin ^{2} x\) can be found (a) by squaring the Maclaurin series for sin x or (b) by using the identity \(\sin ^{2} x=(1-\cos 2 x) / 2\).

Alternative means By comparing the first four terms, show that the Maclaurin series for \(\cos ^{2} x\) can be found (a) by squaring the Maclaurin series for cos x or (b) by using the identity \(\cos ^{2} x=(1+\cos 2 x) / 2\).

Patterns in coefficients Find the next two terms of the following Taylor series. \(\sqrt{1+x}: \quad 1+\frac{1}{2} x-\frac{1}{2 \cdot 4} x^{2}+\frac{1 \cdot 3}{2 \cdot 4 \cdot 6} x^{3}-\cdots\)

Patterns in coefficients Find the next two terms of the following Taylor series. \(\frac{1}{\sqrt{1+x}}: \quad 1-\frac{1}{2} x+\frac{1 \cdot 3}{2 \cdot 4} x^{2}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} x^{3}+\cdots\).

Designer series Find a power series that has (2,6) as an interval of convergence.

Composition of series Use composition of Taylor series to find the first three terms of the Maclaurin series for the following functions. a. \(e^{\sin x}\) b. \(e^{\tan x}\) c. \(\sqrt{1+\sin ^{2} x}\)

Approximations Choose a Taylor series and a center point a to approximate the following quantities with an accuracy of at least \(10^{-4}\). \(\cos 40^{\circ}\)

Approximations Choose a Taylor series and a center point a to approximate the following quantities with an accuracy of at least \(10^{-4}\). \(\sin (0.98 \pi)\)

Approximations Choose a Taylor series and a center point a to approximate the following quantities with an accuracy of at least \(10^{-4}\). \(\sqrt[3]{83}\)

Different approximation strategies Suppose you want to approximate \(\sqrt[3]{128}\) to within \(10^{-4}\) of the exact value. a. Use a Taylor polynomial centered at 0. b. Use a Taylor polynomial centered at 125. c. Compare the two approaches. Are they equivalent?

Approximations Choose a Taylor series and a center point a to approximate the following quantities with an accuracy of at least \(10^{-4}\). \(1 / \sqrt[4]{17}\)

Mean Value Theorem Explain why the Mean Value Theorem (Theorem 4.9 of Section 4.6) is a special case of Taylor's Theorem.

Version of the Second Derivative Test Assume that f has atleast two continuous derivatives on an interval containing a with \(f^{\prime}(a)=0\). Use Taylor's Theorem to prove the following version of the Second Derivative Test: a. If \(f^{\prime \prime}(x)>0\) on some interval containing a, then f has a local minimum at a. b. If \(f^{\prime \prime}(x)<0\) on some interval containing a, then f has a local maximum at a.