Thursday, 3 March 2016

calculus - Evaluate this power series



Evaluate the sum



x+23x3+2345x5+234567x7+



Totally no idea. I think this series may related to the sinx series because of those missing even powers. Another way of writing this series:




k=0(2k)!!(2k+1)!!x2k+1.


Answer



In this answer, I mention this identity, which can be proven by repeated integration by parts:
π/20sin2k+1(x)dx=2k2k+12k22k123=12k+14k(2kk)


Your sum can be rewritten as
f(x)=k=01(2k+1)4k(2kk)x2k+1


Combining (1) and (2), we get
f(x)=π/20k=0sin2k+1(t)x2k+1dt=π/20xsin(t)dt1x2sin2(t)=π/20dxcos(t)1x2+x2cos2(t)=11x2tan1(xcos(t)1x2)]π/20=11x2tan1(x1x2)=sin1(x)1x2






Radius of Convergence



This doesn't appear to be part of the question, but since some other answers have touched on it, I might as well add something regarding it.



A corollary of Cauchy's Integral Formula is that the radius of convergence of a complex analytic function is the distance from the center of the power series expansion to the nearest singularity. The nearest singularity of f(z) to z=0 is z=1. Thus, the radius of convergence is 1.



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