Inequality(II)
Show that1)(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)] 本帖最後由 -終場ソ使者- 於 30-1-2011 22:11 編輯
1) By AM-GM inequality
n\sqrt{\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
http://upload.wikimedia.org/math/d/5/7/d5758ab5c2c39a8c9371366b97443155.png
http://upload.wikimedia.org/math/a/4/b/a4bef94a463ba3ac12865092e88c4950.png
http://upload.wikimedia.org/math/2/a/e/2ae75d4ad818b69045ba89cf764bf9e9.png
These two basic questions always appear in many reference books.
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