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Thread: Bounds of a summative/product series

  1. #1 Bounds of a summative/product series 
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    Is there a way to evaluate summation/production of a series where the lower bound is greater than the upper bound?

    I know this sounds like a nonsense question ... but this idea just seems like there's more to the concept I'm not aware of. Like "Well, actually you can *whatever*". Either that, or you can have the pleasure of blatantly rectifying my speculation with a "No. It's pure and utter nonsense".


    I'm fully aware of the real reasons why 0! is 1 but I'm wondering if this definition has any truth to it, although I'm pretty sure an extension to non natural (by ) integers does not exist.


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    Forum Professor wallaby's Avatar
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    Depends on what the terms being multiplied or added together are. If they are real numbers then there's no problem, if they were matrices however then you can't change the order of the multiplications. Being specific i'd say that the term of the series must commute under the operation being performed for you to be able to change the order in which you evaluate the series.

    As for extending the factorial (function?) to non-natural numbers, it's not strictly speaking an extension but the Gamma Function has the same properties as the factorial.


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    Thanks wallaby, a few questions.

    Quote Originally Posted by wallaby View Post
    Depends on what the terms being multiplied or added together are. If they are real numbers then there's no problem
    Okay. How would I go about doing that? It doesn't really make sense to me. Can I sum the expansion of for with the binomial theorem?

    (function?)
    The factorial is a function , or as I often see in computer applications. Which is why there's speculation of a possible "arcfactorial" (which btw can't be done). Some think of it as an operation, like addition and subtraction, but as you can see it's just a unique function.

    As for extending the factorial (function?) to non-natural numbers, it's not strictly speaking an extension but the Gamma Function has the same properties as the factorial.
    I'm aware of that (not that I really understand the Gamma function though). But as I asking about the lower and upper bounds of a series, I was wondering if the definition

    held any truth for 0, where the upper bound is less than the lower one. As for the negative integers, of course, they would be undefined, so there's no point in that anyway (argh, my bad for mentioning it in the first place).
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    Forum Professor wallaby's Avatar
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    Looks like i misunderstood the question, as i often do, lets see if i can do better this time.
    Quote Originally Posted by brody View Post
    Okay. How would I go about doing that? It doesn't really make sense to me. Can I sum the expansion of for with the binomial theorem?
    Ok so i presume you are referring to the following expansion? (or some other similar form of the binomial expansion)


    in which case substituting values of n < 0 will not yield an accurate result. Instead you would have to take note that for n = -m < 0
    where m > 0.

    So i guess the expansion can then be written as,


    Basically it's not about whether the upper bound is larger or smaller than the lower bound it's a case of adapting the expression to accept values of 'n' outside of the range of values for which the expression was originally defined.


    Quote Originally Posted by brody View Post
    I'm aware of that (not that I really understand the Gamma function though). But as I asking about the lower and upper bounds of a series, I was wondering if the definition

    held any truth for 0, where the upper bound is less than the lower one. As for the negative integers, of course, they would be undefined, so there's no point in that anyway (argh, my bad for mentioning it in the first place).
    Again i don't think there's an issue with the upper bound being lower than the lower bound when you're just taking the product of real numbers, for this particular example however i think that 0! = 1 from this definition just for convenience.
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