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I evaluated

Integrate[(1 + x/n)^n*Exp[-x], {x, 0, Infinity}] . 

thinking the answer should be approximately $\sqrt{\pi n/2}$. Mathematica gave me

ConditionalExpression[E^n n ExpIntegralE[-n, n], Re[n] > 0 || n ∉ Reals]

which I am not sure how to interpret. How can I get Mathematica to give me something closer to what I was expecting?

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2  
Try Integrate[(1 + x/n)^n*Exp[-x], {x, 0, Infinity} , Assumptions -> {n \[Element] Integers, n > 0}], i.e. use the assumptions. –  b.gatessucks Mar 11 '13 at 14:56

1 Answer 1

up vote 4 down vote accepted

Try

Assuming[Element[n, Integers] && n > 0,
 FunctionExpand@Integrate[(1 + x/n)^n Exp[-x], {x, 0, Infinity}]]

(* E^n n^-n Gamma[1 + n, n] *)
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Thanks very much. To get what I was hoping for I would need Mathematica to be able to give detailed asymptotics for Gamma[1 + n, n] which is too much for it I think –  Majid Mar 11 '13 at 17:20

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