....nice calculus...=
Titu′s lemma::
for any positive numbers :
a_1 ,a_2 ,...,a_n , b_1 ,b_2 ,...,b_n
we have:
(((a_1 +...+a_n )^2 )/(b_1 +...+b_n ))≤(a_1 ^2 /b_1 ) +...+(a_n ^2 /b_n )
proof :
put : x=(x_1 ,...,x_n )∈R^n
:y=(y_1 ,...,y_n )∈R^n
(x.y)^2 ≤∣x∣^2 ∣y∣^2 (cauchy−schwarz inequality)
(x_1 y_1 +...+x_n y_n )^2 ≤(x_(1 ) ^2 +...+x_n ^2 )(y_1 ^2 +...+y_(n ) ^2 )
by applying subsitution :
x_i =(a_i /( (√b_i ))) , y_i =(√b_i ) (i=1,2 ,...,n)
((a_(1 ) ^2 +...+a_(n ) ^2 )/(b_2 +...+b_n ))≤(a_1 ^2 /b_1 )+...+(a_n ^2 /b_n ) ✓✓
|