Find the limit (fairly difficult, may be related to Maclaurin series of e^x)?
Find the given limit:
lim { e^-n * (1 / 0! + n / 1! + n^2 / 2! + n^3 / 3! + ... + n^n / n!) }
as n goes to infinity.
This was a problem on a friend's exam a while back. I ended up evaluating the limit after he and I talked about it, but I had to use some decently high-level tools on the way. If anyone is interested, I'll give the sketch of what I did, but I'm curious to see if anyone can solve it differently.
Both my friend and I thought it was 1 at first. The problem is that the series for e^x doesn't converge uniformly, so you can't just use the convergence of the Maclaurin series. The limit, for those who are still interested, is 1/2 (I debated putting it in the question to begin with...).
It may very well be unsolvable using just ordinary calculus tools. I would still welcome an elementary-ish solution, but mostly I'm just wondering if there's a different proof from my own.
alwbsok: You forgot the (n+1)st derivative term in Taylor's theorem. If you were to fix this, then the resulting limit you would want to show (for the error bound) is
n^(n+1) / (n+1)! - 0,
which isn't true anymore.
Thanks for the serious answer (it's odd how rare they are on here sometimes!).
Waiting to see if there's another proof!
Of course, I meant
"n^(n+1) / (n+1)! - 0"
with the arrow for the limit... hurrah for the inability to correct typos in questions!
@_@ - Okay, now Y!A is just messing with me. I chalked it up to my being tired that I missed the arrow the first time, but the second I know I put it there... -_-.
gianlino: I do agree your limit is equivalent, but I wasn't able to find how to show it...
And since you asked, my proof goes something like this:
If X_1, ... , X_n are independent Poisson random variables with mean 1, then the sum
S_n = X_1 + ... + X_n
is a Poisson random variable with mean n.
Then the limit we want is the limit of
P( S_n
That's bizarre... it keeps cutting out random parts of my details...
The limit is the limit of
P( S_n less than or equal to n ),
which by the central limit theorem is
P( Z less than or equal to 0 ) = 1/2
(where Z is a std normal r.v.)