How can I simplify this summation problem?

[rad0786]

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 [tex]n=24[/itex] is pretty tedious...<br /> <br /> Anybody know if there is a simpler way to compute this?[/tex]

 

[0rthodontist]

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

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.

 

[rad0786]

Yea that's what I meant...

 

[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.

 

[rad0786]

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

 

[Hurkyl]

What's the derivative of the following function?

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

 

[rad0786]

Im sorry Latex is difficult to write in...

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

Is that correct?

 

[Hurkyl]

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

 

[rad0786]

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

 

[Hurkyl]

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

 

[rad0786]

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

 

[shmoe]

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

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)

 

[rad0786]

ohhh okay... now this is making a lot more sense...wow...