Find Power Series and interval of convergence

Question

Find Power Series and interval of convergence of g(x) = x^-2 ln(1+x²)

Answer 1

ln(1+z) =  Σn=1,∞ (-1)^n-1 zⁿ/n     so

 

g(x) = ∑n=1,∞ (-1)^n-1 x^2(n-1)/n          

 

To get an idea about the radius of curvature, consider what happens for purely imaginary values of x,that is  let  x = i b     for real b.

 

g(i b) =  ∑n=1,∞  b^2(n-1)/n         for b = 1 this series diverges logarithmically.  for b >1 the divergence is even faster. 

 

So  the radius of curvature is   |x| <1