Am-gm (2 Viewers)

[Archman]

warning, following post should be classified as extracurricular, so it's ok not to get what's going on.

yes rearrangement:
(norm showed this)
suppose a_1 to a_n >= 0
let x_1=a_1/GM, x_2=a_1*a_2/GM^2, ... x_n=a_1*a_2*...*a_n/GM^n=1

now x_1/x_n + x_2/x_1 + x_3/x_2 +...+ x_n/x_n-1 >= x_1/x_1 + x_2/x_2 +...
a_1/GM + a_2/GM +...+ a_n/GM >= n
so AM >= GM

well at the end of the day, i guess you do prove rearrangement by perturbation, but still this is different and quite nice.

 

[turtle_2468]

nice...

 

[Mill]

I assume when you say "Jensen's" you are referring to his inequality?

I would be interested to see this proof if anyone would care to type it up.

 

[Trefoil]

http://planetmath.org/?op=getobj&from=objects&id=3060