Prove that for each natural it's

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- February 20th 2009, 07:59 PMKrizalidNice identity with natural log. and floor function
Prove that for each natural it's

- February 20th 2009, 08:42 PMTwistedOne151Induction
Use proof by induction

(I): Let n=2

Then

and on 1<x<2, , so the above integral is

.

(II): Assume that for n.

Now,

.

Using the fact that for n<x<n+1 in the rightmost integral,

And we have our proof.

--Kevin C. - February 20th 2009, 08:49 PMkalagota
...

i was about to give the same proof.. but i can't fix the latex.. haha - February 21st 2009, 04:37 AMKrizalid
Okay!

We have hence finally

And we're done! - February 21st 2009, 06:10 PMAjreb
brilliant solution!!