hmm... ratio test is tricky to use on this one. lim ((n+1)!/(n+1)^(n+1))/(n!/n^n) = lim (n+1) * n^n / (n+1)^(n+1) = lim n^n / (n+1)^n. L'Hopital it to hell and back and you get lim n!/n!(n+1) or something. That last step is wrong, I'm sure. But it comes out to something less than one, and the ratio test thus confirms that it converges. [ 03-31-2004: Message edited by: Waisz ]
The ratio test gets you to...
lim{n->inf}(1/(1+n))^n then you can't go any further as far as I know. At least that is what I recall. I may be mistaken. [ 03-31-2004: Message edited by: Naimah ]
Using the nth root test on the original sum, you do get lim (n!)^(1/n)/n = lim 1/n = 0, I think. n^1/n = 1, n-1^1/n = 1, etc so the top would all be 1s. [ 03-31-2004: Message edited by: Waisz ]
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quote:
Gunslinger Moogle wrote this then went back to looking for porn:
If all goes according to plan, I will never ever understand what you guys are talking about, or even have to pretend to.
Basic calculus? I'd be surprised if anyone got through a 4-year college without taking some calc.
code:
Ratio test:
lim (n+1)! * n^n
n->Inf ----------- ----
(n+1)^(n+1) n!
Cancelling the n!, we find:lim (n+1) * n^n
n->Inf -----------
(n+1)*(n+1)^nCancelling the (n+1)'s we find:
lim n^n
n->Inf -------
(n+1)^nUsing distributive property of exponents, we find:
lim [n/(n+1)]^n
n->Inf
Which can be rewritten as:lim 1/[(n+1)/n]^n
n->InfThe limit of the denominator as n approaches infinity is the definition of e.
So, it becomes:
1/eSince e>1, 1/e<1. Therefore, by the ratio test, the series converges
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And I was all like 'Oh yeah?' and Waisz was all like:
Basic calculus? I'd be surprised if anyone got through a 4-year college without taking some calc.
You can if you don't go to school on either coast. I'm glad I took it, though I wish I remembered half of it. 
quote:
Vernaltemptress enlisted the help of an infinite number of monkeys to write:
You can if you don't go to school on either coast. I'm glad I took it, though I wish I remembered half of it.
Ah, well, I stand corrected. Calculus isn't really too bad, but for all the math hating types, being forced to take math sucks, I suppose, so oh well.
