Calculate: lim_{n -> oo) e^(-n) sum_(k=0)^n (n^k)/(k!)

Roger Smith

Roger Smith

Answered question


Calculate: limne-nk=0nnkk!

Answer & Explanation

Terry Ray

Terry Ray

Beginner2022-01-17Added 50 answers

Lakisha Archer

Lakisha Archer

Beginner2022-01-18Added 39 answers

The sum is related to the partial exponential sum, and thus to the incomplete gamma function,
since en(x)=k=0xkkexΓn+1,xΓ(n+1). But
The first term in the asymptotic expansion for Γ(n+1,n)=ndttnet
The higher order terms are in principle straightforward to compute. Using Stirlings


Expert2022-01-24Added 556 answers

On this page there is a nice collection of evidence. I add another proof which also uses the Stirling formula. enk=0nnkk!=enk=0nkk(nk)nkk!(nk)! limnenk=1n1ekenk2πk(1+O(1/k))2π(nk)(1+O(1/(nk))) limn12π1nk=1n11kn(1kn)=12π01dxx(1x)=Γ(12)22πΓ(1)=12

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