Hi !

Prove that :

:D

Printable View

- Apr 6th 2009, 09:35 AMMooA series limit
Hi !

Prove that :

:D - Apr 6th 2009, 10:10 AMNonCommAlg

i think the limit originally comes from probability, something that i'm a complete idiot at it! (Rofl) 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. - Apr 6th 2009, 10:29 AMMoo
Yes, that was using a probability proof :D

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 :

But

Hence

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 (Tongueout)

Now, if you have time, I'd be interested in seeing this proof you have that uses calculus ^^ - Apr 6th 2009, 11:25 AMNonCommAlg
sure, if i basically can find the solution i had because it goes back to quite long time ago when i was a calculus I student! haha