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Two interesting relationships.\[\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty\frac1{k^k}\\\left(\sum_{k=1}^nk\right)^2=\sum_{k=1}^nk^3\]
 one year ago
 one year ago
Two interesting relationships.\[\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty\frac1{k^k}\\\left(\sum_{k=1}^nk\right)^2=\sum_{k=1}^nk^3\]
 one year ago
 one year ago

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badreferencesBest ResponseYou've already chosen the best response.0
I'm working on proving them... it'll happen, one day.
 one year ago

badreferencesBest ResponseYou've already chosen the best response.0
I'm posting it here so you guys can prove it alongside me. Right now, don't answer it thoughI want to figure it out on my own.
 one year ago

mukushlaBest ResponseYou've already chosen the best response.2
and there is a similar integral \[\int_{0}^{1} x^x \text{d}x=\sum_{k=1}^{\infty } \frac{(1)^{k+1}}{k^k}\]
 one year ago

badreferencesBest ResponseYou've already chosen the best response.0
Oh lawd, I'm still behind on the first proof. There was no need to give me another funny identity.
 one year ago

TuringTestBest ResponseYou've already chosen the best response.3
use the definition of the integral of a function f(x) over an interval [a,b] \[\int_a^bf(x)dx=\lim_{n\to\infty}\sum_{i=1}^n f(a+i\Delta x)\Delta x~~~\text{ where }~~~\Delta x=\frac{ba}n\]and the first one sort of answers itself for the other use induction
 one year ago

mukushlaBest ResponseYou've already chosen the best response.2
second one is a really interesting identity... and the short and neat answer to this question : Show that for any given positive integer \(n\) there are \(n\) distinct positive integers such that product of them is a complete cube and sum of them is a complete square.
 one year ago

TuringTestBest ResponseYou've already chosen the best response.3
I am looking for a really awesome visual proof of that identity I saw once, I hope I find it.
 one year ago

mukushlaBest ResponseYou've already chosen the best response.2
Exper gave me a link about triangles i cant remember...santosh what was it?
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
lol ... i forgot ... what was that related to?
 one year ago

mukushlaBest ResponseYou've already chosen the best response.2
i cant remember \[1^2+2^2+3^2+...+n^2=?\]or\[1^3+2^3+3^3+...+n^3=?\]
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
oh ... that was just visual proof of ... http://mathoverflow.net/questions/8846/proofswithoutwords
 one year ago

TuringTestBest ResponseYou've already chosen the best response.3
http://mathoverflow.net/questions/8846/proofswithoutwords about halfway down this page, the image with the colored square is the proof I was referring to
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
http://www.math.com/tables/expansion/power.htm
 one year ago

TuringTestBest ResponseYou've already chosen the best response.3
that page is not as pretty
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
dw:1345404870788:dw
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
nice geometry ... i never thought they would add up to make a single square again.
 one year ago

badreferencesBest ResponseYou've already chosen the best response.0
I got the second one before the first... I guess I'm moving onto the identity @mukushla posted.
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
I got pretty stuck up evaluating this one .. dw:1345406507509:dw I don't understand why Mathematica or Maple doesn't give it's value.
 one year ago

mukushlaBest ResponseYou've already chosen the best response.2
is it 0 to \(\infty\)?
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
ah yes ... i tried few days back.
 one year ago

mukushlaBest ResponseYou've already chosen the best response.2
well what was the result then?
 one year ago

experimentXBest ResponseYou've already chosen the best response.0
no result ... not even approximation. it's pretty obvious where it converges http://www.wolframalpha.com/input/?i=integrate+1%2Fx^x+from+0+to+infinity
 one year ago
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