15. Power Series

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

The interval of convergence of the power series ∑ cₖ(x−3)ᵏ could be (−2,8).

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₀∞ (-x/10)²ᵏ

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₁∞ (xᵏ/kᵏ)

In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.

∑ (from n = 0 to ∞) [(x + 1)²ⁿ] / 9ⁿ

Is ∑ₖ₌₀ ∞ (5x − 20)ᵏ a power series? If so, find the center a of the power series and state a formula for the coefficients cₖ of the power series.

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

Radius of convergence Find the radius of convergence for the following power series.

∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ

Intervals of Convergence

Intervals of Convergence

In Exercises 1–36, for what values of x does the series converge (b) absolutely?

∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₁∞ sinᵏ(1/k) xᵏ

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.

∑ₖ₌₁∞ (kx)ᵏ

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) = 2/(1 − 2x)² using f(x) = 1/(1 − 2x)

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence.

∑ (from n = 1 to ∞) [ (√(n + 1) − √n)(x − 3)ⁿ ]

Radius of convergence Find the radius of convergence for the following power series.

∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)

Representing functions by power series Identify the functions represented by the following power series.

∑ₖ₌₀∞ (xᵏ)/(2ᵏ)