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Problem 11.3.67a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function f(x) = √x has a Taylor series centered at 0.
Problem 11.3.67b
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
b. The function f(x) = csc x has a Taylor series centered at π/2.
Problem 11.3.67d
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. If p(x) is the Taylor series for f centered at 0, then p(x−1) is the Taylor series for f centered at 1.
Problem 11.3.35
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.
ln (1 + x²)
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 11.3.41
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 + x⁴)⁻¹
Problem 11.3.43
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.
sinh x²
Problem 11.3.9b
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.
b. Write the power series using summation notation.
f(x) = 1/x², a=1
Problem 11.3.71
Any method
a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. You do not need to use the definition of the Taylor series coefficients.
b. Determine the radius of convergence of the series.
f(x) = (1 + x²)⁻²/³
Problem 11.3.11a
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) = e⁻ˣ, a=0
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 11.3.13a
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)=2/(1−x)³, a=0
Problem 11.3.15a
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) = (1 + x²)⁻¹, a = 0
Problem 11.3.17a
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) = e²ˣ, 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 11.3.15b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = (1 + x²)⁻¹, a = 0
Problem 11.3.13b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=2/(1−x)³, a=0
Problem 11.3.19b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = tan ⁻¹ (x/2), a = 0
Problem 11.3.9c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = 1/x², a=1
Problem 11.3.11c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = e⁻ˣ, a=0
Problem 11.3.13c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x)=2/(1−x)³, a=0
Problem 11.3.17c
Taylor series and interval of convergence
c. Determine the interval of convergence of the series.
f(x) = e²ˣ, a = 0
Problem 11.3.21b
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=3ˣ, a=0
Problem 11.3.1
How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?
Problem 11.3.5
Suppose you know the Maclaurin series for f and that it converges to f(x) for |x|<1. How do you find the Maclaurin series for f(x²) and where does it converge?
Problem 11.3.7
In terms of the remainder, what does it mean for a Taylor series for a function f to converge to f?
Problem 11.3.79
{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.
f(x) = ∜x with a=16; approximate ∜13.
Problem 11.3.77
{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.
f(x) =∛x with a=64; approximate ∛60.
Problem 11.3.29a
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) = 1/x, a = 1
Problem 11.3.31a
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) = ln x, a = 3