Inequality(II)

數學小子

Show that
1)(n!)^3/n<n(n+1)²/4  (where integer n>1)
2)1/1²+1/2²+1/3²+^+1/n²>=6n/[(n+1)(2n+1)]

-終場ソ使者-

1) By AM-GM inequality
n\sqrt[n]{\prod_{k=1}^n k^3}" src="http://upload.wikimedia.org/math/0/6/d/06d1d9aa1157a205ac2da37994d8b565.png">
({n!})^{\frac{3}{n}}" src="http://upload.wikimedia.org/math/3/7/c/37cfc3503372672f00d33ad35dd964fa.png">
(n!)^\frac{3}{n}" src="http://upload.wikimedia.org/math/5/0/d/50df4c48627d45201e210a1bc0504dfe.png">
2) Apply Cauchy-Schwarz inequality




These two basic questions always appear in many reference books.