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badreferences

  • 3 years ago

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

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  1. badreferences
    • 3 years ago
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    I'm working on proving them... it'll happen, one day.

  2. mukushla
    • 3 years ago
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    ver nice integral

  3. badreferences
    • 3 years ago
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    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.

  4. mukushla
    • 3 years ago
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    and there is a similar integral \[\int_{0}^{1} x^x \text{d}x=\sum_{k=1}^{\infty } \frac{(-1)^{k+1}}{k^k}\]

  5. badreferences
    • 3 years ago
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    Oh lawd, I'm still behind on the first proof. There was no need to give me another funny identity.

  6. mukushla
    • 3 years ago
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    ;)

  7. TuringTest
    • 3 years ago
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    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

  8. mukushla
    • 3 years ago
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    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.

  9. TuringTest
    • 3 years ago
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    I am looking for a really awesome visual proof of that identity I saw once, I hope I find it.

  10. mukushla
    • 3 years ago
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    Exper gave me a link about triangles i cant remember...santosh what was it?

  11. experimentX
    • 3 years ago
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    lol ... i forgot ... what was that related to?

  12. mukushla
    • 3 years ago
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    i cant remember \[1^2+2^2+3^2+...+n^2=?\]or\[1^3+2^3+3^3+...+n^3=?\]

  13. experimentX
    • 3 years ago
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    oh ... that was just visual proof of ... http://mathoverflow.net/questions/8846/proofs-without-words

  14. TuringTest
    • 3 years ago
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    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

  15. TuringTest
    • 3 years ago
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    oh same link lol

  16. experimentX
    • 3 years ago
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    http://www.math.com/tables/expansion/power.htm

  17. TuringTest
    • 3 years ago
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    that page is not as pretty

  18. experimentX
    • 3 years ago
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    |dw:1345404870788:dw|

  19. experimentX
    • 3 years ago
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    Interesting!!

  20. TuringTest
    • 3 years ago
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    indeed :D

  21. mukushla
    • 3 years ago
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    yeah :D

  22. experimentX
    • 3 years ago
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    nice geometry ... i never thought they would add up to make a single square again.

  23. badreferences
    • 3 years ago
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    I got the second one before the first... I guess I'm moving onto the identity @mukushla posted.

  24. experimentX
    • 3 years ago
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    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.

  25. mukushla
    • 3 years ago
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    is it 0 to \(\infty\)?

  26. experimentX
    • 3 years ago
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    ah yes ... i tried few days back.

  27. mukushla
    • 3 years ago
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    well what was the result then?

  28. experimentX
    • 3 years ago
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    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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