15. Power Series

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 ∞) [ (ln n) xⁿ ]

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.

f. Diverges for x = 4.9

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = 2x/(1 + x²)²

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.

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 ]

Find the power series representation centered at $x=0$ of the following function. Give the interval of convergence for the resulting series. 

$f\left(x\right)=\frac{1}{1-x^3}$

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

∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = ln √(4 − x²)

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).

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = ln √(1 − x²)

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

∑ₖ₌₀∞ (xᵏ)/(2ᵏ)