<p>It was sarcastic [or irony ;)]...
once we randomly find any solution, we will know that this one is right.
I am trying to solve it using linear change of denominator 3,5,7,9,11...
and by average.</p>
<p>i could write down the proof for why that it is with the fourth dimension.<br>
1/1^2+1/2^2+1/3^2...... approaches pi^2/6, i would have to play around with it to know why.</p>
<p>
[quote]
It was sarcastic
[/quote]
Oh sorry I thought you were serious. I didnt know brainy ppl had half a sense of humor :p</p>
<p>3+5+7+9...... all you do is the average, (because the median is the same) to figure out the sum, they all have the same difference.</p>
<p>1^2+2^2+3^2+4^2+5^2.......x^2</p>
<p>all you have to do is look at increasing three digit numbers, and the amount of combinations in an increasing three digit number is 1+2+3+4+5+6+7+(1+2+3+4+5+6)+(1+2+3+4+5).......</p>
<p>but there is any easier way to express this, it could be 9(8)(7)/6 because an increasing number is only one out of six.</p>
<p>now look 100(1)+/99(2)/+98(3)+/97(4)/+96(5)+/95(6)/.........+/1(100)/= 102(101)(100)/6</p>
<p>and 99(1)+97(2)+95(3)+93(5)...............+1(50)</p>
<p>so the terms in / / are going to be repeated again, and you divide that by two so you get the series below, and you get (102)(101)(100)/24 as the sum which is (2n+2)(2n+1)(2n)/24 which is equal to the sum of the nth squares.</p>
<p>jpsi, the linearity of the denominator won't prove anything, just because it doesnt have a wacky sign around it doesnt mean its linear. The whole function is not linear, as we see by the graph. 1/x isnt linear.</p>
<p>What does 'wacky' mean?</p>
<p>I said that change of denominator is linear. 1/x is not. but (x+1)(x+1)-xx is:
3 for x = 1, 5 for x = 2 etc.</p>
<p>tongos, good job, but you found something else. :p You found the sum of 1² + 2² + 3² .... + x².</p>
<p>I asked the sum of:
(1/1²) + (1/2²) + (1/3²) ..... + (1/x²)</p>
<p>:)</p>
<p>Hey randomgr, i posted a solution for the problem with continuity:
<a href="http://talk.collegeconfidential.com/showpost.php?p=160701&postcount=52%5B/url%5D">http://talk.collegeconfidential.com/showpost.php?p=160701&postcount=52</a></p>
<p>Heh. MS, i have a nice problem that arised when I was thinking about yours [I cannot concentrate on immunology because of this task! I think I will finally sit ad start to write it on paper. ;)]</p>
<p>My task is:</p>
<p>Find such an equation, that when it is raised to x-th power, it will give: (n over x), [I meant newton symbol].</p>
<p>OR </p>
<p>specifically to your task it will be:
((n over 1)(n over n-3)+(n over 2)(n over n-4) and so on)/(n!)^2</p>
<p>And now...
n over 1 [x1], n over 2 [x2], can be shown as: (1+S)^n , where S is the equation that raised to x will give (n over n-x-1)... and is dependant only on n.
[which will be probably n<em>coefficient</em>average... I cannot escape from this average!]</p>
<p>MS, are you sure you're not making fun of us :p ??? (well, i'm sure u are...:D)</p>
<p>i mean, as far as i know, there's no closed formula, which will depend only on n, for<br>
S<em>n=1/1+1/4+1/9+... (\sum</em>{i=1}^n \frac 1{k^2})</p>
<p>There are two ways of proving that lim S_n=pi^2/6:</p>
<ol>
<li>use trigo to place the finite sum between 2 sums which, when n->oo, have the same limit, pi^2/6</li>
</ol>
<p>OR</p>
<ol>
<li>consider f(x)=1/x^2, and write S<em>n as \integral</em>1^n f(x)dx less THE REMINDER... which will come out nicely... </li>
</ol>
<p>so, you are making fun of us, aren't you? :)</p>
<p>I'm not kidding you.
[quote]
There are two ways of proving that lim S<em>n=pi^2/6
[/quote]
Limit as x approaches what? ∞?
[quote]
consider f(x)=1/x^2, and write S</em>n as \integral_1^n f(x)dx
[/quote]
Well you are basically integrating 1/x² which is not my question. I'm asking for a discrete sum (x = positive integers) rather than an integral. If you integrate wrt x, from 1 to n, n will assume all values in between, say, 1 and 2...</p>
<p>same with method #1</p>
<p>well, i'm not kidding you either!!</p>
<p>i know what you want, a closed formula for
1/1+1/(2^2)+1/(3^2)+...+1/(n^2), where n is a positive natural number </p>
<p>Everything is pink and cute, except that there is no closed formula!! hmmm, at least i think zoo.. i mast meik some calculaishions...</p>
<p>or if there is, plzplzplz tell me tell me tell me...</p>
<p>Thank you:) <em>smiles showing bright teeth and blinking many times</em></p>
<p>oh, and mercurysquad:</p>
<p>**<em>gives you cotton candy</em><a href="zante">/b</a></p>
<p>:D:D:D</p>
<p>Hehe pavalon. Relax. I know there ain't a closed form sum for that series :p</p>
<p>My physics teacher asked me this question in 11th grade when he got tired of me answering all his questions... I couldn't solve it then coz I hadn't studied progressions or calc :)</p>
<p>so you where making fun of us... :D:D</p>
<p>i know how to make an aprox for S_n, that's what i wanted to say with integrals...
the REMNDER is actually a nice function, maybe i'll post it, if i have the will-power:D</p>
<p>Pavalon: I didn't quite catch you solution what are those "/alpha"s?</p>
<p>Finally I found the damn solution =)):
users.tellas.gr/~randomgr/pics/solution.gif</p>
<p>I hope it's legible...</p>