## 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

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

2. mukushla

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.

4. mukushla

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.

6. mukushla

;)

7. 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

8. mukushla

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

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

10. mukushla

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

11. experimentX

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

12. mukushla

i cant remember $1^2+2^2+3^2+...+n^2=?$or$1^3+2^3+3^3+...+n^3=?$

13. experimentX

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

14. 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

15. TuringTest

16. experimentX
17. TuringTest

that page is not as pretty

18. experimentX

|dw:1345404870788:dw|

19. TuringTest
20. experimentX

Interesting!!

21. TuringTest

indeed :D

22. mukushla

yeah :D

23. experimentX

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.

25. 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.

26. mukushla

is it 0 to $$\infty$$?

27. experimentX

ah yes ... i tried few days back.

28. mukushla

well what was the result then?

29. 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