A sum I wish I never came across!

1. Feb 1, 2010

Apteronotus

I've come across the following summation

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

moreover, as $$\Delta t \rightarrow 0, x\rightarrow 1^-$$

Does the sum converge? to what?

My thoughts....
The sum as $$n \rightarrow \infty$$ is simply the Mclaren series of $$(1-x)^{-1}$$, so as $$x \rightarrow 1^-$$, the sum should diverge to $$+ \infty$$, however, we have the $$\Delta t$$ in the front that $$\rightarrow 0$$, and thats as far as my intellect takes me...
any ideas?

Last edited: Feb 1, 2010
2. Feb 1, 2010

elibj123

dt doesn't seem to participate in the expression, did you forget it somewhere?

3. Feb 1, 2010

Apteronotus

Yes, I'm so sorry. Its fixed now.

4. Feb 1, 2010

hamster143

You have three variables, n, x, and $\Delta t$.

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

If there are some interrelations between them, that's a different story.

5. Feb 1, 2010

dacruick

if x < 1 it converges.