BackDifferentiating 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)
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)
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)
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)
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)
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)
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.
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?
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?
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
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
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
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
{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.