Proving the Integral Bound for I_n: 0 < I_n < 1/(n+1)

[lordy2010]

Homework Statement

Let I_{n} = \int_{0}^{1}x^{n}e^{-x}dx

Show that 0 < I_{n} < \frac{1}{n+1}

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.

 

[Dick]

x^n*e^(-x) is less than x^n on [0,1], isn't it?

 

[lordy2010]

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!