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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.
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.
