1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook