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Problem 11.2.15
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (x/3)ᵏ
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 11.2.20
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (2x)ᵏ/k!
Problem 11.2.23
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₁∞ ((−1)ᵏ⁺¹(x−1)ᵏ)/k
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 11.2.28
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)
Problem 11.2.21
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯
Problem 11.2.44
Combining power series Use the geometric series
f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,
to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.
f(x³) = 1/(1 − x³)
Problem 11.2.53
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) = − 1/(1 + x)² using f(x) = 1/(1 + x)
Problem 11.2.49
Combining power series Use the power series representation
f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,
to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.
p(x) = 2x⁶ ln(1 − x)
Problem 11.2.47
Combining power series Use the power series representation
f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,
to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.
f(3x) = ln (1 − 3x)
Problem 11.2.51
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) = 2/(1 − 2x)² using f(x) = 1/(1 − 2x)
Problem 11.2.42
Combining power series Use the geometric series
f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,
to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.
g(x) = x³/(1 − x)
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 11.2.63d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
If f(x)=∑ₖ₌₀∞ cₖ xᵏ=0, for all x on an interval (−a, a), then cₖ = 0, for all k.
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 11.2.65
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 11.2.66
A useful substitution Replace x with x−1 in the series ln (1+x) = ∑ₖ₌₁∞ ((−1)ᵏ⁺¹ xᵏ)/k to obtain a power series for ln x centered at x = 1. What is the interval of convergence for the new power series?
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 11.2.73
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) = e⁻³ˣ
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 11.2.78
Inverse sine Given the power series
1/√(1 − x²) = 1 + (1/2)x² + (1 ⋅ 3)/(2 ⋅ 4) x⁴ + (1 ⋅ 3 ⋅ 5)/(2 ⋅ 4 ⋅ 6) x⁶ +⋯
for −1<x<1, find the power series for f(x) = sin ⁻¹ x centered at 0.
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 11.2.76
{Use of Tech} Remainders Let
f(x) = ∑ₖ₌₀∞ xᵏ = 1/(1−x) and Sₙ(x) = ∑ₖ₌₀ⁿ⁻¹ xᵏ
The remainder in truncating the power series after n terms is Rₙ = f(x) − Sₙ(x), which depends on x.
a. Show that Rₙ(x) = xⁿ /(1−x).
b. Graph the remainder function on the interval |x| < 1, for n=1, 2, and 3 . Discuss and interpret the graph. Where on the interval is |Rₙ(x)| largest? Smallest?
c. For fixed n, minimize |Rₙ(x)| with respect to x. Does the result agree with the observations in part (b)?
d. Let N(x) be the number of terms required to reduce |Rₙ(x)| to less than 10⁻⁶. Graph the function N(x) on the interval |x|<1.
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 11.3.3
Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.
Problem 11.3.67e
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
e. The Taylor series for an even function centered at 0 has only even powers of x.
Problem 11.3.21a
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)=3ˣ, a=0
Problem 11.3.33b
Taylor series
b. Write the power series using summation notation.
f(x) = 2ˣ, a = 1
Problem 11.3.29b
Taylor series
b. Write the power series using summation notation.
f(x) = 1/x, a = 1