# Sequence math homework help

1. Jul 3, 2006

Hi... I am working on a problem...

$$41.25 \sum_{n=0}^\24 \frac{n}{x^n}$$

(on the top of the Sigma, it should say 24, NOT 4)

I am searching, but can't seem to find a way to reduce that.

Computing that up to $$n=24[/itex] is pretty tedious... Anybody know if there is a simpler way to compute this? Last edited: Jul 3, 2006 2. Jul 3, 2006 ### 0rthodontist You mean [tex]41.25 \sum_{n=0}^{24} \frac{n}{x^n}$$

I'm pretty sure this can be simplified by a generating function, step 1 is to get x only in positive powers by factoring out x-24.

3. Jul 3, 2006

Yea that's what I meant...

4. Jul 3, 2006

### 0rthodontist

The g.f. you are looking for generates 24, 23, 22, 21, ... 3, 2, 1, 0, 0, 0, ... (you can see that, right?)
So step 1 is to find the g.f. for the sequence 0, -1, -2, -3, etc.
Step 2 is to find the g.f. for the sequence 24, 24, 24 and then add it to the g.f. from step 1 to get 24, 23, ... 1, 0, -1, -2, -3, ...
Step 3 is to find the g.f. that generates 25 0's and then generates 1, 2, 3, .... Then add it to the summed g.f. from step 2 (to get rid of the negative terms in that g.f.) and you have the function you want.

5. Jul 3, 2006

Thank you Orthodontist... yea i can see your method it looks good!
Thanks!

6. Jul 3, 2006

### Hurkyl

Staff Emeritus
What's the derivative of the following function?

$$f(x) = \sum_{n = 0}^{24} x^{-n}$$

7. Jul 3, 2006

Im sorry Latex is difficult to write in...

but f'(x) = sum(n=1,24) -nx^(-n-1)

Is that correct?

8. Jul 3, 2006

### Hurkyl

Staff Emeritus
Yep. And that looks an awful lot like the sum you wanted. (And, to boot, you already know how to compute my sum!)

9. Jul 3, 2006

Im sorry... I don't see how that makes it any easier?

10. Jul 3, 2006

### Hurkyl

Staff Emeritus
Because you already know a simpler expression for $f(x) = \sum_{n = 0}^{24} x^{-n}$

11. Jul 3, 2006

O I am so confused right now....

I was actually looking for a FAST way to compute $$41.25 \sum_{n=0}^{24} \frac{n}{x^n}$$

(it is 24 on top of the sigma, not 4)

I'm confused because I don't know what taking the first derivative of $$f(x) = \sum_{n = 0}^{24} x^{-n}$$ had to do with anything

12. Jul 3, 2006

### shmoe

Take the derivative, then modify it to look like the sum you are after.

You have a simple expression for f(x), it's a geometric series, so you can write it in a form without the summation. Follow the same steps (differentiate, etc.) and you will have the sum you are after in a a nice closed form (no summation)

13. Jul 3, 2006