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Problem 11.4.3
How would you approximate e⁻⁰ᐧ⁶ using the Taylor series for eˣ?
Problem 11.4.4
Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 0.
Problem 11.4.37
Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.∫₀⁰ᐧ²⁵ e⁻ˣ² dx
Problem 11.4.46
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
Problem 11.4.50
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.
tan ⁻¹ (1/2)
Problem 11.4.47
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.
cos 2
Problem 11.4.41
{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ³⁵ tan ⁻¹x dx
Problem 11.4.44
{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ² (ln (1 + t))/t dt
Problem 11.4.38
{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ² sin x² dx
Problem 11.4.25
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
f(x) = eˣ
Problem 11.4.30
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
f(x) = (1 − x)⁻¹
Problem 11.4.27
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
f(x) = ln (1 + x)
Problem 11.4.84b
Symmetry
b. Use infinite series to show that sin x is an odd function. That is, show sin (-x) = -sin x.
Problem 11.4.75
Probability: tossing for a head The expected (average) number of tosses of a fair coin required to obtain the first head is ∑ₖ₌₁∞ k(1/2)ᵏ. Evaluate this series and determine the expected number of tosses. (Hint: Differentiate a geometric series.)
Problem 11.4.76a
Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.
a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.
Problem 11.4.66
Limits with a parameter Use Taylor series to evaluate the following limits. Express the result in terms of the nonzero real parameter(s).
lim ₓ→₀ (eᵃˣ − 1)/x
Problem 11.4.51
Evaluating an infinite series Let f(x) = (eˣ − 1)/x, for x ≠ 0, and f(0)=1. Use the Taylor series for f centered at 0 to evaluate f(1) and to find the value of ∑ₖ₌₀∞ 1/(k+1)!
Problem 11.4.54
Evaluating an infinite series Write the Maclaurin series for f(x) = ln (1+x) and find the interval of convergence. Evaluate f(−1/2) to find the value of ∑ₖ₌₁∞ 1/(k 2ᵏ)
Problem 11.4.73
Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.
f(x) = ∫₀ˣ sin t² dt
Problem 11.4.71
Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.
f(x) = eᶜᵒˢ ˣ
Problem 11.4.65c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ∑ₖ₌₀∞ (ln 2)ᵏ/k! = 2
Problem 11.4.79a
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
a. Compute S′(x) and C′(x).
Problem 11.4.79b
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.
Problem 11.4.79c
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
c. Use the polynomials in part (b) to approximate S(0.05) and C(−0.25).
Problem 11.4.79d
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 10⁻⁴?
Problem 11.4.79e
{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
e. How many terms of the Maclaurin series are required to approximate C(−0.25) with an error no greater than 10⁻⁶?
Problem 11.4.78a
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
a. Expand the integrand in a Taylor series centered at 0.
Problem 11.4.78b
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
b. Integrate the series to find a Taylor series for Si.
Problem 11.4.78c
Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.
c. Approximate Si(0.5) and Si(1). Use enough terms of the series so the error in the approximation does not exceed 10⁻³.
Problem 11.4.83
L'Hôpital's Rule by Taylor series Suppose f and g have Taylor series about the point a.
a. If f(a) = g(a) = 0 and g′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent withl’Hôpital’s Rule.
b. If f(a) = g(a) =f′(a) = g′(a) = 0 and g′′(a) ≠ 0, evaluate lim ₓ→ₐ f(x)/g(x) by expanding f and g in their Taylor series. Show that the result is consistent with two applications of 1'Hôpital's Rule.