Suppose ln(1 + 4x) is expressed as a power series, centered at zero. What is the radius of convergence?

Question

Suppose ln(1 + 4x) is expressed as a power series, centered at zero. What is the radius of convergence?

Answer 1

ln (1 + 4x ) = ∫ [4/(1+4x)] d x = ∫ {4 [1−4x + 16x²−64x³ + 256x⁴ −1024x⁵ + ⋅⋅⋅ ] } d x =

= [ 4x−(4²x²/2 ) + ( 4³x³/3 )−( 4⁴x⁴/ 4) +( 4⁵x⁵/5 ) − (4⁶x⁶ / 6 ) +⋅⋅⋅]

=∑_n=1∞(-1)ⁿ⁺¹(4ⁿ/ n ) xⁿ

Apply the Ratio Test

Lim|( a_n+1) /( a_n ) |= Lim | {4n/(n +1)} x | = |4x| <1

that is |x| <1/4

-1/4 < x < 1/4