Open study

is now brainly

With Brainly you can:

  • Get homework help from millions of students and moderators
  • Learn how to solve problems with step-by-step explanations
  • Share your knowledge and earn points by helping other students
  • Learn anywhere, anytime with the Brainly app!

A community for students.

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

See more answers at
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

Join Brainly to access

this expert answer


To see the expert answer you'll need to create a free account at Brainly

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

Not the answer you are looking for?

Search for more explanations.

Ask your own question

Other answers:

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.
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
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.
I am looking for a really awesome visual proof of that identity I saw once, I hope I find it.
Exper gave me a link about triangles i cant remember...santosh what was it?
lol ... i forgot ... what was that related to?
i cant remember \[1^2+2^2+3^2+...+n^2=?\]or\[1^3+2^3+3^3+...+n^3=?\]
oh ... that was just visual proof of ... about halfway down this page, the image with the colored square is the proof I was referring to
oh same link lol
that page is not as pretty
indeed :D
yeah :D
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.
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.
is it 0 to \(\infty\)?
ah yes ... i tried few days back.
well what was the result then?
no result ... not even approximation. it's pretty obvious where it converges^x+from+0+to+infinity

Not the answer you are looking for?

Search for more explanations.

Ask your own question