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#1 2006-02-25 10:04:09
- mikau
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who woulda thunk it?
The limit of the nth root of n as n approaches infinity equals.... 1!!! I don't know about you guys but I find this extremely remarkable!
The proof of this uses just a simple common technique. I'll use the method of finding the limit of ln (n^(1/n)) as n approaches infinity, and raising e to the power of that limit.
limit of ln (n^(1/n)) as n approaches infinity
= limit of 1/n ln( n) as n approaches infinity = limit of (ln n)/n as n approaches infinity ( = ∞/∞) since this is an indeterminant form, we use L' hopitals rule, 1/n / 1 = 1/n. As n approaches infinity 1/n = 0.
e^0 = 1 so the limit of the the nth root of n as n approaches infinity is one! Thats just awsome! Check it on a calculator if you don't believe me.
Last edited by mikau (2006-02-25 10:06:24)
A logarithm is just a misspelled algorithm.
#2 2006-02-25 10:20:38
- mikau
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Re: who woulda thunk it?
This reminds me of e in some ways. This seems to suggest 1^∞ = ∞ but this is impossible. Therefore it must be something like the limit of (1 + 1/x )^n as n approaches infinity and x is some function of n, and n does not equal x. (otherwise this would be e). I wonder if we could figure out the relation between n and x...
A logarithm is just a misspelled algorithm.
#3 2006-02-25 10:56:37
- ryos
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Re: who woulda thunk it?
Given that 1/∞ = 0, it's not that surprising at all...
El que pega primero pega dos veces.
#4 2006-02-25 11:15:39
- mikau
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Re: who woulda thunk it?
Yes and no. I mean, if you have 1/n, when n becomes significantly large, the exponant tends to be very small, but the base tends to get very large. Just like the definition of e. limit of (1 + 1/x)^x as x approaches infinity. You're inclined to say 1/x = 0 and eliminate it, but the infinitely small difference, to the infinitieth power makes a difference. Likewise, I was inclined to believe the smallness of the exponant would be "canceled out" by the hugeness of the base.
A logarithm is just a misspelled algorithm.
#5 2006-02-25 12:34:09
- John E. Franklin
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Re: who woulda thunk it?
I haven't tried to understand your explanations, but it isn't remarkable to me.
#*#=2
#*#*#=3
#*#*#*#=4
#*#*#*#*#=5
#*#*#*#*#*#=6
The #'s go up slowly, but multiplying is so powerful, why wouldn't you expect to
approach one from above? I would.
'Course I can't remember L'Hopital's Rule by its name. I have a lot to review.
Don't explain for me, I'll do some research...
Imagine for a moment that even an earthworm may possess a love of self and a love of others.
#6 2006-02-25 16:04:44
- mikau
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Re: who woulda thunk it?
maybe you guys are more familiar with limits then me and it doesn't come to such a suprise.
At any rate, the reason I didn't conclude that from the start, I could have looked at the exponant 1/n and figured this comes to zero so its one no matter what, but singling out and isolating a piece of a limit and evaluating can be dangerous, ESPECIALLY when they are directly connected to infinitly large numbers. I repeat: In the definition of e: Limit of (1 + 1/n)^n as n approaches infinity. Your inclined to notice the 1/n term and rewrite it as zero, but this infinitly small fraction is raised to an infinitly large power! So it does make a difference! Here we have the reverse, an infinitly large number being raised to an infinitly small power. For all I knew it might have ended up being 1/e or something like that. Thats why I made no assumptions.
Like I said, when infinitly small and infinitly large come in contact, sometimes they "cancel eachother out" so isolating each individual limit can be dangerous. Don't you agree?
Last edited by mikau (2006-02-25 16:40:58)
A logarithm is just a misspelled algorithm.
#7 2006-02-26 04:54:40
- John E. Franklin
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Re: who woulda thunk it?
.99 ^ big number, goes down.
1.01^ big number, goes up.
Imagine for a moment that even an earthworm may possess a love of self and a love of others.
#8 2006-02-26 05:12:30
- Ricky
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Re: who woulda thunk it?
I'm not really suprised at 1, but I really like the way mikau took the limit. Never would have thought of doing that.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
#9 2006-02-26 07:22:47
- mikau
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Re: who woulda thunk it?
you mean taking the natural log, finding the limit, and raising e to that power? Thats a pretty common technique in my mathbook.
Anyways, the title of this thread is "who woulda thunk it?" clearly the answer is "everyone woulda thunk it.... except mikau..."
I feel stupid...
Last edited by mikau (2006-02-26 07:29:18)
A logarithm is just a misspelled algorithm.
#10 2006-02-26 08:46:56
- mathsyperson
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Re: who woulda thunk it?
Wait a minute...
It was shown somewhere else that this equation:
Had the solution of x = the nth root of n.
Surely, what you've just said means that:
Why did the vector cross the road?
It wanted to be normal.
#11 2006-02-27 05:07:55
- mikau
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Re: who woulda thunk it?
now I'm confused.
technically the limit I found seems to suggest 1^∞ = ∞ but this is obviously not true. This is just the limit. 1 to any power = 1 no matter what. The nth root of n as n approaches infinity, tends to be 1, but 1 is the limit, it can never equal 1.
A logarithm is just a misspelled algorithm.
#12 2006-02-27 07:35:50
- MathsIsFun
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Re: who woulda thunk it?
And, as John said, anything below 1 goes to 0.
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman