Difference between revisions of "User:Temperal/The Problem Solver's Resource8"
(→Diverging-Converging Theorem: sum) |
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Given real numbers <math>a_1 \ge a_2 \ge ... \ge a_n \ge 0</math> and <math>b_1 \ge b_2 \ge ... \ge b_n</math>, we have | Given real numbers <math>a_1 \ge a_2 \ge ... \ge a_n \ge 0</math> and <math>b_1 \ge b_2 \ge ... \ge b_n</math>, we have | ||
− | + | <math>{\frac{\sum a_ib_i}{n}} \ge {\frac{\sum a_i}{n}}{\frac{\sum b_i}{n}}</math>. | |
===Minkowsky's Inequality=== | ===Minkowsky's Inequality=== |
Revision as of 17:33, 9 October 2007
Intermediate Number TheoryThese are more complex number theory theorems that may turn up on the USAMO or Pre-Olympiad tests. This will also cover diverging and converging series, and other such calculus-related topics. General Mean InequalityTake a set of functions Note that
I Chebyshev's InequalityGiven real numbers
Minkowsky's InequalityGiven real numbers
Nesbitt's InequalityFor all positive real numbers
Schur's inequalityGiven positive real numbers
Fermat-Euler IdentitityIf Gauss's TheoremIf Power Mean InequalityFor a real number
Diverging-Converging TheoremA series ErrataAll quadratic residues are |