Thursday, May 26, 2016

analysis - Summing the series (1)kfrac(2k)!!(2k+1)!!a2k+1




How does one sum the series S=a23a3+2435a5246357a7+



This was asked to me by a high school student, and I am embarrassed that I couldn't solve it. Can anyone give me a hint?!


Answer



HINT (a2+1)S=1aS by transmuting the coefficient recurrence to a differential equation.



1=(a2+1)S+aS=f(fS) for f=(a2+1)1/2



S=f1f1=sinh1(a)(a2+1)1/2


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