15. Power Series

Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.

e. Diverges for x = 8

 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?

Intervals of Convergence

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

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

Assume that the series ∑ aₙ(x − 2)ⁿ converges for x = −1 and diverges for x = 6. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.

a. Converges absolutely for x = 1

Theory and Examples

Suppose that a₁, a₂, a₃, …, aₙ are positive numbers satisfying the following conditions:

i) a₁ ≥ a₂ ≥ a₃ ≥ …;

ii) the series a₂ + a₄ + a₈ + a₁₆ + … diverges.

Show that the series

a₁/1 + a₂/2 + a₃/3 + …

diverges.

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

Intervals of Convergence

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

∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]

Intervals of Convergence

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

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

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

∑ₖ₌₀∞ k(x−1)ᵏ

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

∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹

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)

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

∑ₖ₌₀∞ (k²⁰ xᵏ)/(2k+1)!

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

x +x³/3 +x⁵/5 +x⁷/7 + ...

Intervals of Convergence

In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

∑ (from n = 1 to ∞) [ (x − 1)ⁿ / √n ]

Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.

∞

Σ (x/9)³ᵏ

k = 0