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A sum I wish I never came across!

  1. Feb 1, 2010 #1
    I've come across the following summation

    [tex]lim_{\stackrel{\Delta t \rightarrow 0}{n \rightarrow \infty}}\left(\Delta t \sum_{k=0}^n x^k\right)[/tex]

    moreover, as [tex]\Delta t \rightarrow 0, x\rightarrow 1^-[/tex]

    Does the sum converge? to what?

    My thoughts....
    The sum as [tex]n \rightarrow \infty [/tex] is simply the Mclaren series of [tex](1-x)^{-1}[/tex], so as [tex]x \rightarrow 1^- [/tex], the sum should diverge to [tex]+ \infty[/tex], however, we have the [tex]\Delta t[/tex] in the front that [tex] \rightarrow 0[/tex], and thats as far as my intellect takes me...
    any ideas?
     
    Last edited: Feb 1, 2010
  2. jcsd
  3. Feb 1, 2010 #2
    dt doesn't seem to participate in the expression, did you forget it somewhere?
     
  4. Feb 1, 2010 #3
    Yes, I'm so sorry. Its fixed now.
     
  5. Feb 1, 2010 #4
    You have three variables, n, x, and [itex]\Delta t[/itex].

    If they are all allowed to vary independently, the limit does not exist - you can construct sequences of (n,x,[itex]\Delta t[/itex]) which approach any number you want.

    If there are some interrelations between them, that's a different story.
     
  6. Feb 1, 2010 #5
    if x < 1 it converges.
     
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