Let $a_n=\frac{\Gamma(n+h)}{\Gamma(n+1)\Gamma(h)}$ for $0 I will be thankful for a counter example. To prove this I tried using dominated convergence theorem, but that does not work since I don't know if $b_n$ is tight. I appreciate any hints or ideas to approach this.
asymptotic behavior of a nice summation
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limits
summation
gamma-function