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Problem 11.1.45
Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.
f(x) = sin x, a = π/2
Problem 11.4.22
Limits Evaluate the following limits using Taylor series.
lim ₓ→∞ x(e¹/ˣ − 1)
Problem 11.2.54
Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
g(x) = x/(1 + x²)² using f(x) = 1/(1 + x²)
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 11.1.12
Use of Tech Linear and quadratic approximation
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 a.
c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
f(x)=e⁻²ˣ, a=0; approximate e⁻⁰ᐧ².
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 11.2.26
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ (3x + 2)ᵏ/k
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 11.2.30
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ k(x−1)ᵏ
Problem 11.2.65
Shifting power series If the power series f(x)=∑ cₖ xᵏ has an interval of convergence of |x|<R, what is the interval of convergence of the power series for f(x−a), where a ≠ 0 is a real number?
Problem 11.1.53
{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.
sin x ≈ x − x³/6 on [π/4, π/4]
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 11.2.74
Exponential function In Section 11.3, we show that the power series for the exponential function centered at 0 is
eˣ = ∑ₖ₌₀∞ (xᵏ)/k!, for −∞ < x < ∞
Use the methods of this section to find the power series centered at 0 for the following functions. Give the interval of convergence for the resulting series.
f(x) = x²eˣ
Problem 11.2.69
Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)
∑ₖ₌₀∞ e⁻ᵏˣ
Problem 11.4.7
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − 1)/x
Problem 11.2.67
Series to functions Find the function represented by the following series, and find the interval of convergence of the series. (Not all these series are power series.)
∑ₖ₌₀∞(√x − 2)ᵏ
Problem 11.2.32
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (-x/10)²ᵏ
Problem 11.3.3
Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.
Problem 11.4.33
Differential equations
a. Find a power series for the solution of the following differential equations, subject to the given initial condition
b. Identify the function represented by the power series.
y′(t) − y = 0, y(0) = 2
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 11.2.17
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ (xᵏ/kᵏ)
Problem 11.4.62
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₁∞ (x²ᵏ)/k
Problem 11.1.3
The first three Taylor polynomials for f(x)=√(1+x) centered at 0 are p₀ = 1, p₁ = 1+x/2, and p₂ = 1 + x/2 − x²/8. Find three approximations to √1.1.
Problem 11.1.80
{Use of Tech} Best center point Suppose you wish to approximate cos (π/ 2) using Taylor polynomials. Is the approximation more accurate if you use Taylor polynomials centered at 0 or at π/6? Use a calculator for numerical experiments and check for consistency with Theorem 11.2. Does the answer depend on the order of the polynomial?
Problem 11.2.13
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ sinᵏ(1/k) xᵏ
Problem 11.4.20
Limits Evaluate the following limits using Taylor series.
lim ₓ→₁ (x 1)/(ln x)
Problem 11.3.37
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.
1/(1 − 2x)
Problem 11.4.58
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹
Problem 11.4.69
A limit by Taylor series Use Taylor series to evaluate lim ₓ→₀ ((sin x)/x)¹/ˣ²
Problem 11.2.20
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ/k!