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Infinity in Mathematics (Calculus and Series)

  1. Oct 8, 2007 #1
    Could some one please explain to me 2 things

    1) I have seen integrals that are between 0 and ∞ and also between -∞ and ∞. What does this mean

    2) I have also seen sigma series (∑) between n=1 and ∞. What doe this mean

    Thanks heaps
     
  2. jcsd
  3. Oct 8, 2007 #2

    CompuChip

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    Actually, an expression like
    [tex] \int_0^\infty f(x) \, \mathrm{d}x [/tex]
    is just a shorthand notation for
    [tex] \lim_{a \to \infty} \int_0^a f(x) \, \mathrm{d}x. [/tex]
    Likewise,
    [tex] \int_{-\infty}^\infty f(x) \, \mathrm{d}x = \lim_{a \to -\infty} \int_a^0 f(x) \, \mathrm{d}x + \lim_{b \to \infty} \int_0^b f(x) \, \mathrm{d}x[/tex]
    and
    [tex]\sum_{n = 1}^\infty a_n = \lim_{N \to \infty} \sum_{n = 1}^N a_n.[/tex]
     
  4. Oct 8, 2007 #3
    Thanks CompuChip, but i dont understand, how do they exist, what do they literally mean, could you give a example please.

    Thanks
     
  5. Oct 8, 2007 #4

    matt grime

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    I think we need to find out what you think an integral, and a limit are, since the above seem self explanatory.
     
  6. Oct 8, 2007 #5

    CompuChip

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    For example, suppose we want to evaluate
    [tex]\int_0^\infty e^{-x} \, \mathrm{d}x. [/tex]
    Then we first calculate
    [tex]\int_0^a e^{-x} \, \mathrm{d}x = \left. -e^{-x} \right|_{x = 0}^a = -e^{-a} - (-e^{-0}) = 1 - e^{-a}[/tex].
    Now take the limit:
    [tex]
    \int_0^\infty e^{-x} \, \mathrm{d}x =
    \lim_{a \to \infty} \int_0^a e^{-x} \, \mathrm{d}x =
    \lim_{a \to \infty} 1 - e^{-a} =
    1 - \lim_{a \to \infty} e^{-a} = 1 - 0 = 1.
    [/tex]

    That's all there is to it... which part didn't you understand?
     
  7. Oct 8, 2007 #6
    Thanks, it makes perfect sense now :)
     
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