I found a report that relates the sum of exponentials to a particular expression. They argued that:
$$ \sum_{n=1}^N e^{\frac{-nL}{\lambda}} = \frac{e^{\frac{-L}{\lambda}}-e^{\frac{-NL}{\lambda}}}{1-e^{\frac{-L}{\lambda}}}. $$
As $L\to\infty$, I can see that these approximate each other (from graphing it...), but I can't see how to relate them directly and I can also show that it is not true if $N=1$. Any ideas for why these two terms would be related?
Thanks! (Sorry if this has been discussed, I don't know what it is called so don't know how to look it up...)
$S_n=a + ar + ar^2 + ar^3 +...+ ar^n.$ $S_n= e^u + (e^u)(e^u) + (e^u)(e^u)^2+ ...+ (e^u)^n$
Where $a=e^u, r = e^u $ and $u=\frac {-L}{c}$ Finite sum for a geometric sum is:
$S_n= \frac {a(1-r^n)}{1-r}$ when $ (r < 1)$
Hence
$S_n= \frac {e^u(1-(e^u)^n)}{1-e^u}$.
Set $n = N +1 $
When $N = 1 $ the result equals $0$ which is not equal to $e^R$ where $R = \frac {-L}{c}$.