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

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

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I'm working on proving them... it'll happen, one day.
ver nice integral
I'm posting it here so you guys can prove it alongside me. Right now, don't answer it though--I want to figure it out on my own.

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and there is a similar integral \[\int_{0}^{1} x^x \text{d}x=\sum_{k=1}^{\infty } \frac{(-1)^{k+1}}{k^k}\]
Oh lawd, I'm still behind on the first proof. There was no need to give me another funny identity.
;)
use the definition of the integral of a function f(x) over an interval [a,b] \[\int_a^bf(x)dx=\lim_{n\to\infty}\sum_{i=1}^n f(a+i\Delta x)\Delta x~~~\text{ where }~~~\Delta x=\frac{b-a}n\]and the first one sort of answers itself for the other use induction
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.
I am looking for a really awesome visual proof of that identity I saw once, I hope I find it.
Exper gave me a link about triangles i cant remember...santosh what was it?
lol ... i forgot ... what was that related to?
i cant remember \[1^2+2^2+3^2+...+n^2=?\]or\[1^3+2^3+3^3+...+n^3=?\]
oh ... that was just visual proof of ... http://mathoverflow.net/questions/8846/proofs-without-words
http://mathoverflow.net/questions/8846/proofs-without-words about halfway down this page, the image with the colored square is the proof I was referring to
oh same link lol
http://www.math.com/tables/expansion/power.htm
that page is not as pretty
|dw:1345404870788:dw|
http://math.stackexchange.com/questions/61482/intuitive-explanation-for-the-identity-sum-limits-k-1n-k3-left-sum-l
Interesting!!
indeed :D
yeah :D
nice geometry ... i never thought they would add up to make a single square again.
I got the second one before the first... I guess I'm moving onto the identity @mukushla posted.
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.
is it 0 to \(\infty\)?
ah yes ... i tried few days back.
well what was the result then?
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

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