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

- anonymous

- schrodinger

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- anonymous

I'm working on proving them... it'll happen, one day.

- anonymous

ver nice integral

- anonymous

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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## More answers

- anonymous

and there is a similar integral \[\int_{0}^{1} x^x \text{d}x=\sum_{k=1}^{\infty } \frac{(-1)^{k+1}}{k^k}\]

- anonymous

Oh lawd, I'm still behind on the first proof. There was no need to give me another funny identity.

- anonymous

;)

- TuringTest

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

- anonymous

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.

- TuringTest

I am looking for a really awesome visual proof of that identity I saw once, I hope I find it.

- anonymous

Exper gave me a link about triangles i cant remember...santosh what was it?

- experimentX

lol ... i forgot ... what was that related to?

- anonymous

i cant remember \[1^2+2^2+3^2+...+n^2=?\]or\[1^3+2^3+3^3+...+n^3=?\]

- experimentX

oh ... that was just visual proof of ...
http://mathoverflow.net/questions/8846/proofs-without-words

- TuringTest

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

- TuringTest

oh same link lol

- experimentX

http://www.math.com/tables/expansion/power.htm

- TuringTest

that page is not as pretty

- experimentX

|dw:1345404870788:dw|

- TuringTest

http://math.stackexchange.com/questions/61482/intuitive-explanation-for-the-identity-sum-limits-k-1n-k3-left-sum-l

- experimentX

Interesting!!

- TuringTest

indeed :D

- anonymous

yeah :D

- experimentX

nice geometry ... i never thought they would add up to make a single square again.

- anonymous

I got the second one before the first... I guess I'm moving onto the identity @mukushla posted.

- experimentX

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.

- anonymous

is it 0 to \(\infty\)?

- experimentX

ah yes ... i tried few days back.

- anonymous

well what was the result then?

- experimentX

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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