What is the limit?
lim(n→∞)[1-((n-1)/n)^n]
It is the probability of at least one success from n trials (with replacement) with a 1/n chance of success.
Having just had a quick look at it, I haven't figured it yet. But numerically, it appears to be around 0.632, if it exists.
Knock yourself out Steve & Ben!
Posts
Gods Own County
4 years ago
6 comments:
Ok I'm writing it down on something and I'll have an exact solution after I wake up some more
My mate Rowan has proposed that the solution is 1 - 1/e, but didn't have a proof yet, as of about 4:30 yesterday.
damn smart people maths talk, yo'all think of us yokel type folk in your fancy speaches and what not.
Rowan's solution is correct (as per the msg amy hopefully forwarded to you today) I also can show you a proof.. maybe when im feeling less lazy i'll write one up in paint :P
Continuity of the exp() map and you exp(ln(.)) it, then can use L'hopital's rule easily.
Yep, I got your message Steve, thanks!
I was having a bit of a look yesterday, and was thinking about expanding the binomial to a summation. I didn't really go far from there, but then had a look online and found a similar example (for (1 + 1/n)^n) which can be shown the same way to be limited by exp(1). I didn't continue with the line of reasoning, but figured the negative within the summation would give an alternating sign in the summation, and that sounded familiar in terms of known limits.
I'm interested in what you refer to as the 'continuity of the exp() map', and I suppose this is a different route to the solution? Perhaps if you don't end up getting it into paint (or msWord with eqn editor + print screen) then we can find a back of an envelope next time I see you and fill it with elegant mathematics for which it was never designed!
I do quite like the envelope idea :) I'll show you tonight if you're going to Travs thing.
Post a Comment