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

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badreferences Group TitleBest ResponseYou've already chosen the best response.0
I'm working on proving them... it'll happen, one day.
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
ver nice integral
 2 years ago

badreferences Group TitleBest 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.
 2 years ago

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

badreferences Group TitleBest 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.
 2 years ago

TuringTest Group TitleBest 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
 2 years ago

mukushla Group TitleBest 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.
 2 years ago

TuringTest Group TitleBest 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.
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
Exper gave me a link about triangles i cant remember...santosh what was it?
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
lol ... i forgot ... what was that related to?
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
i cant remember \[1^2+2^2+3^2+...+n^2=?\]or\[1^3+2^3+3^3+...+n^3=?\]
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
oh ... that was just visual proof of ... http://mathoverflow.net/questions/8846/proofswithoutwords
 2 years ago

TuringTest Group TitleBest 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
 2 years ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.3
oh same link lol
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
http://www.math.com/tables/expansion/power.htm
 2 years ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.3
that page is not as pretty
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
dw:1345404870788:dw
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
Interesting!!
 2 years ago

TuringTest Group TitleBest ResponseYou've already chosen the best response.3
indeed :D
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
nice geometry ... i never thought they would add up to make a single square again.
 2 years ago

badreferences Group TitleBest 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.
 2 years ago

experimentX Group TitleBest 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.
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
is it 0 to \(\infty\)?
 2 years ago

experimentX Group TitleBest ResponseYou've already chosen the best response.0
ah yes ... i tried few days back.
 2 years ago

mukushla Group TitleBest ResponseYou've already chosen the best response.2
well what was the result then?
 2 years ago

experimentX Group TitleBest 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
 2 years ago
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