# Difficult series

1. Jul 15, 2004

### Feynman

Good morning ,
I have to calculate these series , Help me

$$\displaystyle\sum_{k=1}^{i-1}C_{i}^{k}(tk)^{k-2}k(i-k)^{i-k+1}$$
Thanks

2. Jul 15, 2004

### Wong

Are you sure it is $$C_{i}^{k}$$? When k is smaller than i, I think $$C_{i}^{k}$$ is 0?

3. Jul 15, 2004

### Feynman

YA I'm sure
so???????????

4. Jul 15, 2004

### Zurtex

I don't really understand the maths involved here, but I think the point was if $C_{i}^{k} = 0$ then your sum is 0 + 0 + 0 + ... + 0.

5. Jul 15, 2004

### Feynman

u write it is $C_{k}^{i}$

6. Jul 15, 2004

### Feynman

So how calculate these serieS?

7. Jul 19, 2004

### Feynman

8. Jul 19, 2004

### Wong

Maybe you should try to use Maple for this particular problem? Some guys told me that Maple can handle this kind of symbolic computations.

9. Jul 19, 2004

### Feynman

Good morning
I'm must proof the calculation step of this series and not the result from maple!!!!

10. Jul 19, 2004

### Wong

Now, your series contains a variable t. Surely if the sum makes sense, it should sum to a polynomial. So, the problem is what do you want to prove? The problem makes sense if the proposition is like "prove that the series sums to a polynomial in t with coefficients of the form..." or like "prove that the series sums to the nth derivative of ... function".

So what exactly is the proposition?

11. Jul 19, 2004

### Feynman

I need the steps calculation of these series!!

12. Jul 19, 2004

### Wong

sorry then....maybe I can't help, as I am unclear about what you want to prove. The expression is a polynomial in t. It may be "simplified" into any form that you may deem suitable.

13. Jul 19, 2004

### Feynman

Can u calculate me this series?

14. Jul 21, 2004

### Feynman

So Wong can u help me?

15. Jul 21, 2004

### Wong

Sorry, maybe I can't help....

16. Jul 22, 2004

### Zurtex

Have you tried expanding it out and trying the first few terms? I'm just doing that now.

But it does look complex, I get the 1st 2 terms as being $(i/t)(i - 1)^i$ and $i(i-1)(i-2)^{i-1}$

Perhaps looking at this wrong, it may be more easy for trying different values of i.

17. Jul 22, 2004

### pnaj

Write out, on paper, the first 5 terms, the kth term and the last two terms. See if you can find a pattern. For example, the first term MIGHT pair with the last term to equal zero, the second might pair with the second to last, etc.

If that doesn't work, try induction on k, see how far you get.

At least have a go!

EDIT: Zurtex ... I didn't notice your post, I'm just pretty much repeating what you said.

Last edited: Jul 22, 2004
18. Jul 22, 2004

### Zurtex

Edit: It's all wrong, sorry.

Last edited: Jul 23, 2004
19. Jul 22, 2004

### Zurtex

Edit: Sorry I tried reducing the seris in a way that was just plain wrong. However it does occur to me that you can take out $i!/t^2$ as a common factor.

Last edited: Jul 23, 2004
20. Jul 26, 2004

### Feynman

Zurtex
THe $i$ is not a complex it is the index!!!!!!!!!!!!!!!!!!!!