General term of the sequence, if it exists

[relinquished™]

Hello. I've been having such a hard time thinking of the general term of this "Sequence". Actually, I'm not even sure if this is a sequence at all, but it looks like it can be simplified into one summation symbol.
<br /> \frac{-2}{6}, \frac{-20}{120}, \frac{-1080}{5040}, \frac{-140400}{362880}, ...<br />

The denominators of every term are actually the factorials of the odd numbers starting from 3, what i can't find is the "pattern" for the numerator.

Thanks for any help :)

 

[damoclark]

There's a nice data base of integer sequences at :

http://www.research.att.com/~njas/sequences/

that you can search.

I played around with the sequence you have given, but couldn't figure anything much out. Do you know anymore of the terms?

 

[relinquished™]

Actually, with a little tinkering I did manage to find the pattern :) I just have one question... It is related to the sequence but its not actually the sequence

is this statement true?

\prod_{n=0} (2n+1) = (1)(3)(5)(7)(9)...

Note: The n in the "Prod" symbol tends to infinity. I don't know how to place an upper limit in the symbol XD

I'm not so familiar with the symbol, i just saw it in the HowToLaTeX FAQ and wondered if it's like the summation symbol (only it means product) :)

Thanks again for that site. It did help me in a way :)

 

[Muzza]

I guess the statement is true, but it doesn't appear to be well-defined...

If you want to know how to place an upper limit:

\prod_{n=0}^{\infty} (2n+1)

 

[relinquished™]

so it should be

\prod_{n=0}}^{\infty} (2n+1) = (1)(3)(5)(7)(9)(11)(13)...


Is it "more" well defined now?

Thanks again

 

[Muzza]

No, it just seems like it's "meaningless" to talk about the product of all odd natural numbers ;)

 

[relinquished™]

well, yeah, it is meaningless. But when it becomes a part of a general term of a series that is a solution to a differential equation it is kinda important :)

which leads me to my last question, (which I know should be part of Differential Equations but my main focus was simplifying the general term of a series) in most differential equations books when I read their solutions they write their general term as (1)(3)(5)(7)...(2n+1) (If the need or occasion arose). My question is if it's more appropriate to write it as

\prod_{n=0}^{\infty} (2n+1)

instead...

Thanks a bunch :)

 

[shmoe]

If your general term is (1)(3)(5)...(2n+1) then you could write it as

\prod_{i=0}^{n}(2i+1)

Note the endpoints carefully. Either one is fine as long as there's no ambiguity for what the ... represent. My preference is towards the \prod notation as long as there are no typsetting issues.

 

[relinquished™]

Thanks for everything :)

 

[Data]

An alternative notation, that is sometimes prettier, and that doesn't involve product notation:

\prod_{i=0}^n (2i+1) = \frac{(2n+1)!}{2^n n!}

Edit: Actually, looking at your situation, this notation might lead to some simplifications too!