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Problem 11.3.31b
Taylor series
b. Write the power series using summation notation.
f(x) = ln x, a = 3
Problem 11.3.33a
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = 2ˣ, a = 1
Problem 11.3.39
Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.
{(eˣ−1)/x if x ≠ 1, 1 if x = 1
Problem 11.3.49a
{Use of Tech} Binomial series
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.
f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.
Problem 11.3.49b
{Use of Tech} Binomial series
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.
Problem 11.3.25a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = ln (x − 2), a = 3
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 11.3.53
Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 11.4 is useful). Use the Maclaurin series
√(1 + x) = 1 + x/2 − x²/8 + x³/16 − ⋯, −1 ≤ x ≤ 1.
√(9 − 9x)
Problem 11.3.57
Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series
(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.
(1 + 4x)⁻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 11.3.25c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = ln (x − 2), a = 3
Problem 11.3.27b
Taylor series
b. Write the power series using summation notation.
f(x)=sin x, a = π/2
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 11.3.27a
Taylor series
a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x)=sin x, a = π/2
Problem 11.3.23a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = cosh 3x, a = 0
Problem 11.3.19a
Taylor series and interval of convergence
a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.
f(x) = tan ⁻¹ (x/2), a = 0
Problem 11.3.61
Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series
(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.
1/(3 + 4x)²
Problem 11.3.65
Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.
f(x) = e⁻ˣ, a = 0
Problem 11.4.58
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹
Problem 11.4.55
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ (xᵏ)/(2ᵏ)
Problem 11.4.62
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k
Problem 11.4.22
Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x(e¹/ˣ − 1)
Problem 11.4.8
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (tan ⁻¹ x − x)/x³"
Problem 11.4.7
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − 1)/x
Problem 11.4.9
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = cosh x, n = 3, a = ln 2
Problem 11.4.11
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − e⁻ˣ)/x
Problem 11.4.20
Limits Evaluate the following limits using Taylor series.
lim ₓ→₁ (x 1)/(ln x)
Problem 11.4.14
Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x sin(1/x)
Problem 11.4.16
Limits Evaluate the following limits using Taylor series.
lim ₓ→₄ (x² 16)/(ln (x 3)}
Problem 11.4.10
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (sin 2x)/x