proposition 7.76 $\pi$ as the circle constant

Rests on

• 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 theorem A.40 Least-upper-bound property ¶
        • 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 ¶
      • 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 definition 7.2 Absolute value ¶ ↺
        • depends_on definition 7.9 Real function ¶
      • 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 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 7.27 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 definition 7.16 Limit ¶ ↺
        • depends_on definition 7.9 Real function ¶ ↺
      • depends_on proposition 7.6 Algebra of limits ¶
        • depends_on definition 7.4 Convergence ¶ ↺
        • depends_on proposition 7.3 Triangle inequality ¶
        • proves proof ch:05-real-analysis@proof-3 ¶
      • proves proof ch:05-real-analysis@proof-11 ¶
    • depends_on proposition 7.48 Ratio test ¶
      • depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      • 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 ¶ ↺
    • 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 definition 13.11 Arc length ¶
  • depends_on definition 13.8 Curve ¶
  • depends_on definition 13.9 Tangent vector to a curve ¶
    • depends_on definition 13.8 Curve ¶ ↺
• 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 corollary A.46 Monotone convergence ¶ ↺
    • depends_on definition 7.45 Series ¶ ↺
    • depends_on proposition 7.6 Algebra of limits ¶ ↺
    • proves proof ch:05-real-analysis@proof-30 ¶
  • depends_on theorem 7.51 Termwise differentiation ¶ ↺
  • proves proof ch:05-real-analysis@proof-46 ¶
• depends_on lemma 7.75 The first quadrant ¶
  • 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 definition 7.11 Sum of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.6 Algebra of limits ¶ ↺
        • proves proof ch:05-real-analysis@proof-13 ¶
      • depends_on theorem 7.34 Rolle ¶
        • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • proves proof ch:05-real-analysis@proof-18 ¶
      • proves proof ch:05-real-analysis@proof-19 ¶
    • proves proof ch:05-real-analysis@proof-20 ¶
  • 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 theorem 7.23 Intermediate value theorem ¶ ↺
• proves proof ch:05-real-analysis@proof-50 ¶

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