15. 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! − ⋯

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

∑ₖ₌₂∞ ((x+3)ᵏ)/(k łn²k)

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.

f(x³) = 1/(1 − x³)