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April 2nd, 2011, 11:33 AM   #1
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Limit

Let be a positive integer, is an arbitrary real number. Find the limit of sequence where:


here the notation is the largest integer that does not exceed .
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April 2nd, 2011, 04:41 PM   #2
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Re: Limit

I think you made a mistake in your equation. I'm not sure what is meant by .a, and you may have forgotten a + sign in the numerator. Do you mean the limit at n approaches infinity?
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April 2nd, 2011, 04:43 PM   #3
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Re: Limit

Many use a decimal to denote multiplication.
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April 2nd, 2011, 04:58 PM   #4
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Re: Limit

Assuming the numerator is actually very quick jottings (at 2 in the morning, so I assume no responsibility for the lack of accuracy in my analysis ) suggest that the limit as is
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April 2nd, 2011, 05:53 PM   #5
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Re: Limit

If that's true, then as n approaches infinity, the limit may be zero.
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April 2nd, 2011, 06:47 PM   #6
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Re: Limit

I was wrong. Doing some research, I found Faulhaber's Formula which can be used to determine the formula for these types of sums in terms of n. Both the numerator and denominator are of degree k+1, therefore the limit at infinity does exist. The exact limit of this problem as n goes to infinity is floor(a)/(k+1) after determining the coefficient of the n^(k+1) term in the numerator.
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April 3rd, 2011, 07:30 AM   #7
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Re: Limit

I don't think so -



Both the left hand and right hand expressions are of the form so the expression in the middle must converge to rather than
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April 3rd, 2011, 10:14 AM   #8
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Re: Limit

Interesting use of the squeeze theorem. I hadn't thought of that.
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