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Proof involving integral

  • Thread starter lordy2010
  • Start date
  • #1
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Homework Statement


Let [tex]I_{n} = \int_{0}^{1}x^{n}e^{-x}dx[/tex]

Show that 0 < [tex]I_{n}[/tex] < [tex]\frac{1}{n+1}[/tex]

Homework Equations


n/a


The Attempt at a Solution


I have been trying to prove this for a long time, and so far I haven't gotten anywhere. I managed to get a reduction formula for it using integration by parts, and I can prove that it is greater than 0, but not that it is less than 1/(n+1). I have tried using induction too, but with no luck.
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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x^n*e^(-x) is less than x^n on [0,1], isn't it?
 
  • #3
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Ah, I can't believe I didn't think of that! I'd tried a bunch of complicated methods, trying to use inverse substition and even writing the antiderivative in terms of a factorial and summation, but it was right in front of me the whole time... Thank you!
 

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