Saturday, 24 August 2019

limits - How to calculate limnrightarrowinftyfracfrac1nnleft(Gamma(n)n+enGamma(n+1,n)right)sqrtn



How can I compute the following limit?



limn1nn(Γ(n)n+enΓ(n+1,n))n




The answer appears to be about 1.25.


Answer



This is the same as the one worked out in this question. We have
Γ(n+1,n)=n!en(nk=0nkk!)n!2

from this question.
Hence, we get that
enΓ(n+1,n)nn+1/2enn!2nn+1/2en2πnn+1/22nn+1/2en=π2

As I have in the comments, the first term can be thrown away. Hence, the limit is π21.25


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