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) = 2x/(1 + x²)²

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)

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 x⁴ᵏ/k²

k = 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.

∞

Σ (x - 1)ᵏ/(k5ᵏ)

k = 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.

∞

Σ (x/9)³ᵏ

k = 0

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 + ...