find a power series representation and radius of convergence for f by integratin

Question

find a power series representation and radius of convergence for f by integrating or differentiating the series for 2/(1+x^2) f(x)=2arctanx

Explanation / Answer

Start with the geometric series 1/(1 - t) = S(n = 0 to 8) t^n for |t| < 1. Let t = -x^2: 1/(1 + x^2) = S(n = 0 to 8) (-1)^n x^(2n) for |-x^2| = |x|^2 < 1 ==> |x| < 1. Multiply both sides by 2x: 2x/(1 + x^2) = S(n = 0 to 8) (-1)^n 2x^(2n+1) for |x| < 1. Integrate both sides from 0 to x: ln(1 + x^2) = S(n = 0 to 8) (-1)^n 2x^(2n+2)/(2n+2) for |x| < 1 (at least). ..................= S(n = 0 to 8) (-1)^n x^(2n+2)/(n+1) ..................= S(k = 1 to 8) (-1)^(k-1) x^(2k)/k, via reindexing. Checking endpoints: x = -1 or 1 ==> S(k = 1 to 8) (-1)^(k-1) / k. This converges by the Alternating Series Test, because {1/k} decreases and converge to 0. So, the interval of convergence is [-1, 1]. I hope this helps!

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