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  • * [http://www.amazon.com/exec/obidos/ASIN/052154677X/artofproblems-20 The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequaliti * [http://www.amazon.com/exec/obidos/ASIN/0486425665/artofproblems-20 Sequences, Combinations, Limits]
    24 KB (3,202 words) - 14:33, 13 January 2025
  • * [[Sequence | Sequences]] and [[Series]] ** [[Cauchy-Schwarz Inequality]]
    2 KB (198 words) - 15:06, 7 December 2024
  • ...h>\lambda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]]. ...isely thus: Let <math>\{ \{a_{ij}\}_{i=1}^n \} _{j=1}^m</math> be several sequences of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a seq
    4 KB (774 words) - 11:12, 29 October 2016
  • ...<math>\bigl( 1/(y+z), 1/(z+x), 1/(x+y) \bigr)</math> are similarly sorted sequences, it follows from the [[Rearrangement Inequality]] that ...nd <math>\bigl(x/(y+z), y/(z+x), z/(x+y)\bigr)</math> are similarly sorted sequences. Then by [[Chebyshev's Inequality]],
    7 KB (1,164 words) - 00:01, 26 July 2024
  • ...s of generality, say these are <math>x</math> and <math>y</math>. Then the sequences <math>(x, 1, 1)</math> and <math>(1, 1, y)</math> are oppositely sorted, yi by [[Chebyshev's Inequality]]. By the [[Cauchy-Schwarz Inequality]] we have
    4 KB (641 words) - 10:19, 26 September 2024
  • ...sqrt{S+6ca}}, \frac{1}{\sqrt{S+6ab}} \right)</math> are similarly oriented sequences. Thus === Alternate Solution using Cauchy ===
    7 KB (1,194 words) - 05:11, 22 October 2024
  • === Proof by Cauchy Induction === We use [[Cauchy Induction]], a variant of induction in which one proves a result for <math>
    14 KB (2,624 words) - 01:50, 11 January 2025
  • ...h>\lambda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]]. ...isely thus: Let <math>\{ \{a_{ij}\}_{i=1}^n \} _{j=1}^m</math> be several sequences of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a seq
    4 KB (764 words) - 17:42, 28 September 2024
  • *''The Cauchy-Schwarz Master Class'' - '''J. Michael Steele'''. ...w=Jedi201708.pdf&rlkey=yo0ip3sg90ehil5ys2mo3mtlo ''Functions, Polynomials, Sequences''] - '''Hojoo Lee'''.
    19 KB (2,581 words) - 13:25, 1 November 2024
  • By the [[Cauchy-Schwarz_Inequality|Cauchy-Schwarz]] inequality, Again by the Cauchy-Schwarz inequality,
    6 KB (1,135 words) - 19:20, 17 June 2022
  • It is a direct consequence of [[Cauchy-Schwarz inequality]]. ...2014})^2}{x_1^2 + x_2^2 + \ldots + x_{2014}^2} \leq N</cmath> for all real sequences <math>x_1, x_2, ..., x_{2014}</math>. Find the sum of the digits of <math>N
    5 KB (761 words) - 19:10, 29 April 2024
  • A <b>Cauchy sequence</b> is defined to be a [[sequence]] <math>x_n</math> such that, fo ...hy sequences in <math>M</math> under the [[equivalence relation]] that two sequences <math>x_n</math> and <math>y_n</math> are equivalent if <math>\lim_{n\to\in
    1 KB (216 words) - 16:53, 29 February 2020
  • ====Cauchy-Swartz Inequality==== ..., such that <math>n</math> is a positive integer, where all members of the sequences are real, then we have:<cmath>(a_1^2+a_2^2+\cdots +a_n^2)(b_1^2+b_2^2+ \cdo
    4 KB (671 words) - 12:59, 22 July 2020
  • ...the real numbers are the construction of the real numbers, convergence of sequences, subsets of the plane as metric spaces, limits, notions of [[continuity]], ...y known constructions of the real numbers are via [[Cauchy sequence|Cauchy sequences]] and [[Dedekind cut|Dedekind cuts]], both of which take <math>\mathbb{Q}</
    9 KB (1,409 words) - 01:41, 30 May 2023
  • Consider infinite sequences <math>\{x_n\}</math> of positive reals such that <math>x_0=1</math> and <ma By Cauchy, the LHS is at least:
    3 KB (440 words) - 22:23, 29 January 2021