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 x∈R. 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)=∫∞ats−1e−tdt,a>0,s∈R. Observe that, for $0
No comments:
Post a Comment