15. Power Series

Suppose a power series converges if |x−3|<4 and diverges if |x−3| ≥ 4. Determine the radius and interval of convergence.

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.

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

∑ₖ₌₀∞ k(x−1)ᵏ

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

∑ₖ₌₀∞ (k²⁰ xᵏ)/(2k+1)!

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

∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)

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

∑ₖ₌₀∞ (-x/10)²ᵏ

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

∑ₖ₌₁∞ (1−cos (1/2ᵏ)) xᵏ

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

∑ₖ₌₀∞ (2x)ᵏ

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

∑ₖ₌₁∞ (kx)ᵏ

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₌₀∞ (x/3)ᵏ

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

∑ₖ₌₁∞ (xᵏ/kᵏ)

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

∑ₖ₌₀∞ (2x)ᵏ/k!

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

∑ₖ₌₁∞ ((−1)ᵏ⁺¹(x−1)ᵏ)/k

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

−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯