Prove that :
i think the limit originally comes from probability, something that i'm a complete idiot at it! there's the idea of the proof in this thread. i know a calculus proof but it's long and very ugly!
so i think it's a good challenge finding a neat calculus (analysis) solution for this question.
Same type as the previous ones :P
Laurent in the thread gave the way to solve it, I'll detail it here
Consider a sequence of independent random variables, following a Poisson distribution with parameter 1 :
Then let's consider
It is easy to prove that it follows a Poisson distribution with parameter n :
We can see that :
Then we must check the conditions for applying the Central Limit Theorem :
And the are independent and identically distributed.
The Central Limit Theorem says that :
converges to the standard normal distribution
In particular, there is a convergence of the cumulative density functions :
, which is because the integrand is even.
This finishes the proof :
Haaa... I really like these kinds of limits that use probability, it's just great
Now, if you have time, I'd be interested in seeing this proof you have that uses calculus ^^