Question

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

Answer 1

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

Answer 2

ver nice integral

Answer 3

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.

Answer 4

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

Answer 5

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

Answer 6

;)

Answer 7

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

Answer 8

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.

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

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

Answer 13

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

Answer 14

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

Answer 15

oh same link lol

Answer 16

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

Answer 17

that page is not as pretty