A sum I wish I never came across

[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 that's as far as my intellect takes me...
any ideas?

 

[elibj123]

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

 

[Apteronotus]

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

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

 

[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.

 

[dacruick]

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 that's as far as my intellect takes me...
any ideas?

if x < 1 it converges.