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Anyone recognize this series expansion?

2,981
2
Anyone recognize this series expansion??

[tex]1+3t+\frac{9t^2}{2!}+\frac{27t^3}{3!}+\frac{51t^4}{4!}+.....[/tex]

I looks kind of like [itex]e^t[/itex] but i am not sure how to deal with it.

Can I factor something... I kind of suck at these. Someone give me a hint.
 

Answers and Replies

nicksauce
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I don't quite see the pattern... how do you get from 27 to 51, and what comes next?
 
2,981
2
I don't quite see the pattern... how do you get from 27 to 51, and what comes next?
Oh...yes that should be an 81

[tex]
1+3t+\frac{9t^2}{2!}+\frac{27t^3}{3!}+\frac{81t^4} {4!}+.....[/tex]
 
nicksauce
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So you have (3t)^0 / 0! + (3t)^1 / 1! + (3t)^2 /2! + (3t)^3 / 3! + ...
Surely you can see what this function is?
 
2,981
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So you have (3t)^0 / 0! + (3t)^1 / 1! + (3t)^2 /2! + (3t)^3 / 3! + ...
Surely you can see what this function is?
I can now! By the way what the Christ is 0! ??? I want to say that for some strange reason it is 1...... but I don't know why??
 
cristo
Staff Emeritus
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I can now! By the way what the Christ is 0! ??? I want to say that for some strange reason it is 1...... but I don't know why??
0!=1, by definition.
 
2,981
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0!=1, by definition.
Yeah. That is what I thought.... though I thought that there was more to it than 'because the math gods said so.'

But I'll take it if that's all there is to it:smile:
 
Dick
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Yeah. That is what I thought.... though I thought that there was more to it than 'because the math gods said so.'

But I'll take it if that's all there is to it:smile:
The math gods said so for a good reason. You want the factorial function to satisfy n!=n*(n-1)!. If you put n=1, then you'd better define 0!=1. You can see you're in big trouble trying to define (-1)!. But that's ok. This is also related to the properties of the gamma function. gamma(n+1)=n!. And gamma(1)=1. gamma(0) is undefined, it's a pole of the gamma function. So we'd better leave (-n)! undefined.
 
HallsofIvy
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Yeah. That is what I thought.... though I thought that there was more to it than 'because the math gods said so.'

But I'll take it if that's all there is to it:smile:
0!= 1 because I SAY SO!!
 

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