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General term of the sequence, if it exists

  1. Mar 13, 2005 #1
    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.
    [tex]
    \frac{-2}{6}, \frac{-20}{120}, \frac{-1080}{5040}, \frac{-140400}{362880}, ...
    [/tex]

    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 :)
     
  2. jcsd
  3. Mar 14, 2005 #2
    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???
     
    Last edited: Mar 14, 2005
  4. Mar 14, 2005 #3
    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?

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

    Note: The n in the "Prod" symbol tends to infinity. I dont 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 :)
     
  5. Mar 14, 2005 #4
    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:

    [tex]\prod_{n=0}^{\infty} (2n+1)[/tex]
     
  6. Mar 14, 2005 #5
    so it should be

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


    Is it "more" well defined now?

    Thanks again
     
  7. Mar 14, 2005 #6
    No, it just seems like it's "meaningless" to talk about the product of all odd natural numbers ;)
     
    Last edited: Mar 14, 2005
  8. Mar 14, 2005 #7
    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

    [tex] \prod_{n=0}^{\infty} (2n+1) [/tex]

    instead...

    Thanks a bunch :)
     
  9. Mar 14, 2005 #8

    shmoe

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    Science Advisor
    Homework Helper

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

    [tex]\prod_{i=0}^{n}(2i+1)[/tex]

    Note the endpoints carefully. Either one is fine as long as there's no ambiguity for what the ... represent. My preference is towards the [itex]\prod[/itex] notation as long as there are no typsetting issues.
     
  10. Mar 14, 2005 #9
    Thanks for everything :)
     
  11. Mar 14, 2005 #10
    An alternative notation, that is sometimes prettier, and that doesn't involve product notation:

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

    Edit: Actually, looking at your situation, this notation might lead to some simplifications too!
     
    Last edited: Mar 14, 2005
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