April 2nd, 2011, 11:33 AM  #1 
Newbie Joined: Apr 2011 Posts: 10 Thanks: 0  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 . 
April 2nd, 2011, 04:41 PM  #2 
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  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?

April 2nd, 2011, 04:43 PM  #3 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Limit
Many use a decimal to denote multiplication.

April 2nd, 2011, 04:58 PM  #4 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  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 
April 2nd, 2011, 05:53 PM  #5 
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  Re: Limit
If that's true, then as n approaches infinity, the limit may be zero.

April 2nd, 2011, 06:47 PM  #6 
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  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.

April 3rd, 2011, 07:30 AM  #7 
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  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 
April 3rd, 2011, 10:14 AM  #8 
Senior Member Joined: Sep 2009 From: Wisconsin, USA Posts: 227 Thanks: 0  Re: Limit
Interesting use of the squeeze theorem. I hadn't thought of that.


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