Τρίτη, 3 Οκτωβρίου 2017

Proof of A.M. $\geq$ G.M. using integration

We like to prove :
Observe that for any  x > 0 :

because, so long as the integrand’s  t  lies strictly increasing or decreasing between  x  and  G , the sign  G – x  and of  1/t – 1/G , must be the same.

Equality sign holds iff  x = G.

Now, replace  x  by    and sum up over  i = 1, 2, … , n.
\ A ³ G