In my class we have the following equation.
$\displaystyle Pr[N(t + ∆t) − N(t) = k] = \frac{e^{\lambda\Delta t} }{ k!}$, where $k = 0, 1, 2, \ldots \quad$ $(1)$
We are talking about the poisson process. I understand what the LHS means of $(1)$ and also what the RHS means.
What I don't understand is the following approximation that is used to prove further results:
$e^{\lambda\Delta t} \approx 1 + \lambda\Delta t + o(\Delta t)$ $\quad (2)$
Could someone explain why this is true?
Thank you for your help!
we have
$$exp(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!}$$
when $x$ is small, the higher order terms vanishes first.
Side remark: is there a typo in the question, should the sign before $\lambda \Delta t$ be positive on the RHS? or should there be a negative in the exponential on the LHS?