I'm taking a course on stochastic analysis. I'm stuck on the very first problem of the lecture notes:
$\lim_{n \to \infty} \left(1+\frac{\lambda}{n} + o(n^{-1})\right)^n = \exp(\lambda)$
Prior to the problem, the lecturer mentioned infinitesimal functions and introduced the Taylor series. I'm not sure how they are useful in proving the above, however.
I thought about taking the log of both sides:
$\lim_{n \to \infty} n\log \left(1+\frac{\lambda}{n} + o(n^{-1})\right) = \lambda$
then substituting $m=n^{-1}$ to get:
$\lim_{m \to 0} m^{-1} \log \left(1+\lambda m + o(m)\right) = \lambda$
I'm lost on where to go next. Does anybody have any ideas?