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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 11.1.73b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
b. Estimate f(0.2) and give a bound on the error in the approximation.
f(x) =√(1+x) ≈ 1 + x/2
Problem 11.3.25b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = ln (x − 2), a = 3
Problem 11.3.27b
Taylor series
b. Write the power series using summation notation.
f(x)=sin x, a = π/2
Problem 11.1.65b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. Let f(x)=x⁵−1 The Taylor polynomial for f of order 10 centered at 0 is f itself.
Problem 11.3.13b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=2/(1−x)³, a=0
Problem 11.3.23b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = cosh 3x, a = 0
Problem 11.3.17b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e²ˣ, a = 0
Problem 11.3.33b
Taylor series
b. Write the power series using summation notation.
f(x) = 2ˣ, a = 1
Problem 11.3.15b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = (1 + x²)⁻¹, a = 0
Problem 11.3.31b
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3
Problem 11.3.19b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = tan ⁻¹ (x/2), a = 0
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 11.3.29b
Taylor series
b. Write the power series using summation notation.
f(x) = 1/x, a = 1
Problem 11.1.71b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
a. Estimate f(0.1) and give a bound on the error in the approximation.
f(x) = tan⁻¹ x ≈ 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 11.3.11b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e⁻ˣ, a=0
Problem 11.3.17c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = e²ˣ, a = 0
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 11.3.9c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = 1/x², a=1
Problem 11.1.65c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. Only even powers of x appear in the nth−order Taylor polynomial for f(x)=√(1+x²) centered at 0.
Problem 11.3.13c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x)=2/(1−x)³, a=0
Problem 11.3.21c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x)=3ˣ, a=0
Problem 11.2.67c
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If f has a Taylor series that converges only on (−2,2), then f(x²) has a Taylor series that also converges only on (−2,2).
Problem 11.3.23c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = cosh 3x, a = 0
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 11.3.11c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = e⁻ˣ, a=0
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 11.3.25c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = ln (x − 2), a = 3
Problem 11.4.81c
{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is
J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ
c. Differentiate J₀ twice and show (by keeping terms through x⁶) that J₀ satisfies the equation x² y′′(x) + xy′(x) + x²y(x)=0.