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Find Power Series and interval of convergence

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

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Richard P. | Fairfax County Tutor for HS Math and ScienceFairfax County Tutor for HS Math and Sci...
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ln(1+z) =  Σn=1,∞ (-1)n-1 zn/n     so
 
g(x) = ∑n=1,∞ (-1)n-1 x2(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,∞  b2(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
 

Comments

At x=±1, the series becomes the alternating harmonic series, which is known to converge (to ln(2)). The interval of convergence is |x| ≤1 .