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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?

  • 3
    As $m$ approaches 0, you can write $\log(1 + x)$ in terms of its taylor expansion.2012-12-27
  • 0
    Where does $x$ come from?2012-12-27
  • 1
    $x$ in this case is $\lambda m + o(m)$.2012-12-27

2 Answers 2

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Write $$A = \lim_{n \to \infty} \left(1+\frac{\lambda}{n} + f(n)\right)^n$$ where $f(n) = o(n^{-1})$.

Then $$\log(A) = n\log\left( \lim_{n \to \infty} \left(1+\frac{\lambda}{n} + f(n)\right)\right)$$

Since $\lim_{n \to \infty} |f(n)| \leq \lim_{n \to \infty}\epsilon/n = 0$. It follows that

$$\log(A) = \log \left( \lim_{n \to \infty} \left(1+\frac{\lambda}{n}\right)^n\right)$$

and therefore:

$$ A = \exp(\lambda)$$

  • 0
    Could you explain the reasoning behind the final step (the "therefore")? It's not obvious to me why $\lim_{n\to\infty}\left(1+\frac{\lambda}{n}\right)^n = \exp(\lambda)$.2012-12-27
  • 0
    This holds by definition. http://en.wikipedia.org/wiki/Exponential_function#Formal_definition2012-12-27
  • 0
    Ah, that's the part I was missing. Thank you.2012-12-27
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You can show and use the following inequality: $$\forall t\geqslant 0,\quad t-\frac{t^2}2\leqslant \log(1+t)\leqslant t.$$ It's classical and actually, it's a bound on the Taylor's remainder. But we can prove it directly.