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        riddles >> putnam exam (pure math) >> Another Limit Problem
        
(Message started by: THUDandBLUNDER on Jan 3^rd, 2007, 5:39pm)

Title: Another Limit Problem
Post by THUDandBLUNDER on Jan 3^rd, 2007, 5:39pm

Title: Re: Another Limit Problem
Post by Barukh on Jan 4^th, 2007, 3:30am

[hide]0.49...[/hide]?

Title: Re: Another Limit Problem
Post by Icarus on Jan 4^th, 2007, 7:28pm

As with the other limit problem, this one can be reduced by means of a Cauchy sum, though it is trickier to pull it off. Subtract out and then add back in (n(n+1)/2) ln n to the numerator. The result can be expressed as

(1/n)(f(1/n) + f(2/n) + ... + f(n/n))/(n ln n) + (n+1)ln n/2n ln n, where f(x) = x ln x

The numerator of the first fraction is the Cauchy sum for f(x) on [0,1], and since f is continuous and bounded on this interval, the integral exists and is finite. Since the denominator is unbounded, the entire fraction goes to zero. The final term obviously has limit 1/2, and therefore so does the entire expression.

Title: Re: Another Limit Problem
Post by Barukh on Jan 5^th, 2007, 4:58am

Alternatively, let F(n) = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif₁ⁿ xln(x) dx. Then, for every finite n, the numerator of the given expression is bounded from above and from below by F(n+1) and F(n).

Title: Re: Another Limit Problem
Post by JP05 on Jan 5^th, 2007, 3:38pm

Question: Barukh, how did you get that integral symbol in that post? If there is supposed to be a toolbar with that symbol I am not seeing it.

Title: Re: Another Limit Problem
Post by THUDandBLUNDER on Jan 5^th, 2007, 4:14pm

on 01/05/07 at 15:38:33, JP05 wrote:

Question: Barukh, how did you get that integral symbol in that post? If there is supposed to be a toolbar with that symbol I am not seeing it.

You can click on the Quote (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_putnam;action=post;num=1167874788;quote=3;title=Post+reply;start=0) link of any post to see how it was written.
There used to be a drop-down menu of maths symbols but it is not functional at the moment.

Title: Re: Another Limit Problem
Post by JP05 on Jan 5^th, 2007, 4:36pm

Oh! I did not know quote would expose what some typed but it does, thanks. Not having a list of named symbol file names make this hard. From the looks of what I see with quote you can include a symbol file from any http source (right?).

Title: Re: Another Limit Problem
Post by balakrishnan on Jan 5^th, 2007, 5:30pm

It is 1/2
Just follows from the fact that the numerator
~ int[x*log(x)]_{x=0 to n}
which is n^2/2*log(n)+O(n^2)
and hence the limit is 1/2

Title: Re: Another Limit Problem
Post by Icarus on Jan 5^th, 2007, 7:38pm

on 01/05/07 at 16:36:48, JP05 wrote:

You can use the image tags [ img ] [ /img ] to post an image from anywhere (these are actually available from one of those buttons above the smileys on the reply page - I suggest going through all the buttons to see what they do. Some of them can be useful). The math symbolry we used to have is still available in the directory http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/. By following that link, you can see the directory contents - though its hard to figure out what the individual gifs are from their names. William had a really nice system set up for using these images as smileys. But unfortunately, he had to upgrade the version of YaBB that the forum runs under, and all his customizations disappeared. While he has restored - and improved - the other customizations, this one never got fixed.

There is a thread in the Suggestions forum (near the top, as it is sticky) on how to display mathematical symbols. If you read the entire thread, you can see a glimpse of the history of this forum. But what is useful starts with this post. (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi?board=riddles_suggestions;action=display;num=1035249563;start=25#41) In particular, in the post after mine, SMQ gives a wonderful script that allows Firefox users to see pages written with the old symbolry in all their glory! If you would like the see the posts I created to start off the 0.999... thread in the Medium forum as they are supposed to look, download Firefox and GreaseMonkey if you haven't already, and install this script. (They were so pretty, and now I can't even redo them as the max size of a post has decreased, and they are too big to modify. :'()