Monday, May 20, 2019

calculus - Evaluating inti0nftydn,fracxn(3n+1)(3n+2)


I'm trying to prove a particular series is convergent, and I would like to use the Cauchy integral test for fun, even though it's not the most convenient. I need to evaluate,


0dnxn(3n+1)(3n+2)


for xR. Using Mathematica I have found the integral to be,


13x2/3[x1/3Γ(0,log(x)/x)Γ(0,2log(x)/x)]


providing x<1. So I just need to express the integral in terms of incomplete gamma functions, but I haven't found a substitution. Can someone offer a hint (and not a complete solution)?


Answer



Hint. Recall that the incomplete gamma function may be defined as Γ(s,a)=ats1etdt,a>0,sR. Observe that, for $0

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