Difference between revisions of "User:Temperal/The Problem Solver's Resource8"
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when <math>k \neq 0</math> and is given by the geometric mean]] of the | when <math>k \neq 0</math> and is given by the geometric mean]] of the | ||
<math>a_i</math> when <math>k = 0</math>. | <math>a_i</math> when <math>k = 0</math>. | ||
+ | |||
+ | ===Diverging-Converging Theorem=== | ||
+ | A series <math>\displaystyle_{i=0}^{\infty}S_i</math> converges iff <math>\displaystyle\lim S_i=0</math>. | ||
===Errata=== | ===Errata=== |
Revision as of 22:13, 5 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 %{\frac{\sum a_ib_i}{n}} \ge {\frac{\sum a_i}{n}}{\frac{\sum b_i}{n}}%. 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 resiues are |