- #1

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If someone could help out by getting me in the right direction or just plain giving me the answer, that would be much appreciated.

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- Thread starter Crazy Gnome
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- #1

- 13

- 0

If someone could help out by getting me in the right direction or just plain giving me the answer, that would be much appreciated.

- #2

nicksauce

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Well probably a best way would be a proof by induction. Do you know what that is?

Edit: I'll add to this. You want to show [tex]\int_{0}^{\infty}t^ne^{-st}dt=\frac{n!}{s^{n+1}}[/tex] for all non-negative integers n. To do this, first show it is true for n=0. Then assume it is true for arbitrary n, and show that it is true for n+1, ie: [tex]\int_{0}^{\infty}t^ne^{-st}dt=\frac{n!}{s^{n+1}}\rightarrow\int_{0}^{\infty}t^{n+1}e^{-st}dt=\frac{(n+1)!}{s^{n+2}}[/tex]. This can easily be done with an integration by parts.

Edit: I'll add to this. You want to show [tex]\int_{0}^{\infty}t^ne^{-st}dt=\frac{n!}{s^{n+1}}[/tex] for all non-negative integers n. To do this, first show it is true for n=0. Then assume it is true for arbitrary n, and show that it is true for n+1, ie: [tex]\int_{0}^{\infty}t^ne^{-st}dt=\frac{n!}{s^{n+1}}\rightarrow\int_{0}^{\infty}t^{n+1}e^{-st}dt=\frac{(n+1)!}{s^{n+2}}[/tex]. This can easily be done with an integration by parts.

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- #3

HallsofIvy

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Crazy Gnome, the step from n to n+ 1 (or k to k+1) should be easy to do using integration by parts. Oh, and, of course, you need to show it is true for n= 1 or n= 0 depending on whether you are talking about all natural numbers or all whole numbers.

- #4

nicksauce

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nicksauce, technical point: you should say :"assume it is true for arbitrary n" because that is what you are trying to prove! What you should say is "assume it is true for a single value of n" and then show it is true for n+1. I personally prefer to say "assume it is true for some k" so as not to confuse the specific value with the general value.

Yes, you're right. I meant to say "an arbitrary n" but left out the "an". And yes, using 'k' is indeed much more clear.

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