## badreferences Group Title Two interesting relationships.$\int_0^1\frac{\mathrm{d}x}{x^x}=\sum_{k=1}^\infty\frac1{k^k}\\\left(\sum_{k=1}^nk\right)^2=\sum_{k=1}^nk^3$ one year ago one year ago

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

2. mukushla Group Title

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 Group Title

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 Group Title

;)

7. TuringTest Group Title

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 Group Title

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 Group Title

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

10. mukushla Group Title

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

11. experimentX Group Title

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

12. mukushla Group Title

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

13. experimentX Group Title

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

14. TuringTest Group Title

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 Group Title

16. experimentX Group Title
17. TuringTest Group Title

that page is not as pretty

18. experimentX Group Title

|dw:1345404870788:dw|

19. TuringTest Group Title
20. experimentX Group Title

Interesting!!

21. TuringTest Group Title

indeed :D

22. mukushla Group Title

yeah :D

23. experimentX Group Title

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 Group Title

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 Group Title

is it 0 to $$\infty$$?

27. experimentX Group Title

ah yes ... i tried few days back.

28. mukushla Group Title

well what was the result then?

29. experimentX Group Title

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