proposition 7.83 Viète's product

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:2398 · p. 236

Rests on

depends_on lemma 7.72 Addition theorems ¶
- depends_on equation 7.53 eq:ana-cos-sin-series ¶
- depends_on equation 5.3 eq:lin-binomial ¶
- depends_on proposition 7.49 Cauchy product ¶
  - depends_on corollary A.46 Monotone convergence ¶
    - depends_on theorem A.40 Least-upper-bound property ¶
      - depends_on definition A.36 Cut ¶
      - depends_on definition A.38 Order on $\R$ ¶
      - proves proof app:A-long-proofs@proof-29 ¶
    - proves proof app:A-long-proofs@proof-33 ¶
  - depends_on definition 7.45 Series ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.4 Convergence ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
  - depends_on proposition 7.6 Algebra of limits ¶
    - depends_on definition 7.4 Convergence ¶ ↺
    - depends_on proposition 7.3 Triangle inequality ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - proves proof ch:05-real-analysis@proof-1 ¶
    - proves proof ch:05-real-analysis@proof-3 ¶
  - proves proof ch:05-real-analysis@proof-30 ¶
- depends_on proposition 7.48 Ratio test ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶ ↺
    - depends_on theorem 7.8 Cauchy criterion ¶
      - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-5 ¶
    - proves proof ch:05-real-analysis@proof-28 ¶
  - depends_on proposition 7.46 Geometric series ¶
    - depends_on corollary A.46 Monotone convergence ¶ ↺
    - depends_on definition 7.45 Series ¶ ↺
    - proves proof ch:05-real-analysis@proof-27 ¶
  - depends_on theorem 7.8 Cauchy criterion ¶ ↺
  - proves proof ch:05-real-analysis@proof-29 ¶
- depends_on proposition 7.6 Algebra of limits ¶ ↺
- proves proof ch:05-real-analysis@proof-47 ¶
depends_on proposition 7.79 The polygon recursion ¶
- depends_on definition 7.74 $\pi$ ¶
  - depends_on axiom 7.1 Completeness of $\R$ ¶
  - depends_on lemma 7.73 $\cos 2 < 0$ ¶
    - depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
    - depends_on proposition 7.52 Alternating series test ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-33 ¶
    - proves proof ch:05-real-analysis@proof-48 ¶
  - depends_on theorem 7.23 Intermediate value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - proves proof ch:05-real-analysis@proof-8 ¶
  - depends_on theorem 7.51 Termwise differentiation ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶ ↺
      - proves proof ch:05-real-analysis@proof-11 ¶
    - depends_on proposition 7.48 Ratio test ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - depends_on theorem 7.50 Power series; radius of convergence ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶ ↺
      - proves proof ch:05-real-analysis@proof-31 ¶
    - proves proof ch:05-real-analysis@proof-32 ¶
- depends_on lemma 7.78 Chord, arc, tangent ¶
  - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    - depends_on theorem 7.35 Mean value theorem ¶
      - depends_on proposition 7.29 Linearity ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-19 ¶
    - proves proof ch:05-real-analysis@proof-20 ¶
  - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
    - depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
    - depends_on equation 5.3 eq:lin-binomial ¶ ↺
    - depends_on proposition 7.49 Cauchy product ¶ ↺
    - depends_on theorem 7.51 Termwise differentiation ¶ ↺
    - proves proof ch:05-real-analysis@proof-46 ¶
  - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
    - depends_on definition 7.74 $\pi$ ¶ ↺
    - depends_on equation 7.58 eq:ana-half-angle ¶
    - depends_on lemma 7.65 The integer part ¶
      - depends_on axiom 3.33 Peano axioms ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on lemma 7.64 Well-ordering of $\N$ ¶
      - proves proof ch:05-real-analysis@proof-42 ¶
    - depends_on lemma 7.75 The first quadrant ¶
      - depends_on corollary 7.36 cor:ana-mvt-consequences ¶ ↺
      - depends_on definition 7.74 $\pi$ ¶ ↺
      - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶ ↺
      - depends_on theorem 7.23 Intermediate value theorem ¶ ↺
      - depends_on theorem 7.51 Termwise differentiation ¶ ↺
      - proves proof ch:05-real-analysis@proof-49 ¶
    - depends_on lemma 7.72 Addition theorems ¶ ↺
    - proves proof ch:05-real-analysis@proof-51 ¶
  - proves proof ch:05-real-analysis@proof-52 ¶
- depends_on lemma 7.72 Addition theorems ¶ ↺
- depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
- proves proof ch:05-real-analysis@proof-53 ¶
depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
depends_on proposition 7.6 Algebra of limits ¶ ↺
proves proof ch:05-real-analysis@proof-55 ¶

Supports

depends_on remark 7.84 One construction, two faces ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Addition theorems	declared	parts/02-mathematical-methods/05-real-analysis.tex:2412
`depends_on`	→	The polygon recursion	declared	parts/02-mathematical-methods/05-real-analysis.tex:2412
`depends_on`	→	Special values, periodicity, and the kernel	declared	parts/02-mathematical-methods/05-real-analysis.tex:2412
`depends_on`	→	Algebra of limits	declared	parts/02-mathematical-methods/05-real-analysis.tex:2412
`depends_on`	←	One construction, two faces	declared	parts/02-mathematical-methods/05-real-analysis.tex:2455
`proves`	←	ch:05-real-analysis@proof-55	declared	parts/02-mathematical-methods/05-real-analysis.tex:2416