Calculus II Power Series Question

Question

Represent the function 5ln(4-x) as a power series (Maclaurin series) in the format: f(x) = ∞∑n=0 c_nxⁿ

1) Find: c₀ , c₁, c₂, c₃ & c₄

2) Also find the radius of convergence, R

Answer 1

1 / (1 - x) = 1 + x + x² + x³ + ... for -1 < x < 1

d/dx[5ln(4 - x)] = -5/(4 - x) = -5 / [4(1 - x/4)] = (-5/4)[1 / (1 - (x/4))]

So, 5ln(4 - x) = (-5/4) ∫ [1 / (1 - (x/4)]dx = (-5/4) ∫ [1 + (x/4) + (x/4)² + (x/4)³ + (x/4)⁴ + ...]dx, for -1 < x/4 < 1

= (-5/4)[x + x²/8 + x³/48 + x⁴/256 + ...] for -4 < x < 4

= (-5/4)x + (-5/32)x² + (-5/192)x³ + (-5/1024)x⁴ + ... for -4 < x < 4

c₀ = 0, c₁ = -5/4, c₂ = -5/32, c₃ = -5/192, c₄ = -5/1024

Interval of convergence = (-4,4). Radius of convergence = (4 - (-4)) / 2 = 4.