anonymous
  • anonymous
Two interesting relationships.\[\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty\frac1{k^k}\\\left(\sum_{k=1}^nk\right)^2=\sum_{k=1}^nk^3\]
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
I'm working on proving them... it'll happen, one day.
anonymous
  • anonymous
ver nice integral
anonymous
  • 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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anonymous
  • 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
  • anonymous
Oh lawd, I'm still behind on the first proof. There was no need to give me another funny identity.
anonymous
  • anonymous
;)
TuringTest
  • 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
  • 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
  • TuringTest
I am looking for a really awesome visual proof of that identity I saw once, I hope I find it.
anonymous
  • anonymous
Exper gave me a link about triangles i cant remember...santosh what was it?
experimentX
  • experimentX
lol ... i forgot ... what was that related to?
anonymous
  • anonymous
i cant remember \[1^2+2^2+3^2+...+n^2=?\]or\[1^3+2^3+3^3+...+n^3=?\]
experimentX
  • experimentX
oh ... that was just visual proof of ... http://mathoverflow.net/questions/8846/proofs-without-words
TuringTest
  • 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
  • TuringTest
oh same link lol
experimentX
  • experimentX
http://www.math.com/tables/expansion/power.htm
TuringTest
  • TuringTest
that page is not as pretty
experimentX
  • experimentX
|dw:1345404870788:dw|
TuringTest
  • TuringTest
http://math.stackexchange.com/questions/61482/intuitive-explanation-for-the-identity-sum-limits-k-1n-k3-left-sum-l
experimentX
  • experimentX
Interesting!!
TuringTest
  • TuringTest
indeed :D
anonymous
  • anonymous
yeah :D
experimentX
  • experimentX
nice geometry ... i never thought they would add up to make a single square again.
anonymous
  • anonymous
I got the second one before the first... I guess I'm moving onto the identity @mukushla posted.
experimentX
  • 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
  • anonymous
is it 0 to \(\infty\)?
experimentX
  • experimentX
ah yes ... i tried few days back.
anonymous
  • anonymous
well what was the result then?
experimentX
  • 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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