proposition 7.46 Geometric series

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:896 · p. 217

Rests on

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 $\R$ ¶
    - depends_on definition A.36 Cut ¶ ↺
  - 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 ¶ ↺
proves proof ch:05-real-analysis@proof-27 ¶

Supports

depends_on lemma 7.86 Terminating or repeating decimals are rational ¶
- depends_on proposition 7.91 The golden ratio ¶
  - depends_on proposition 7.92 Continued fraction and the Fibonacci ratios ¶
    - depends_on remark 7.95 Where $\varphi$ is used ¶
  - depends_on proposition 125.2 Crystallographic restriction ¶
    - depends_on remark 125.3 What the theorem does and does not forbid ¶
  - depends_on remark 7.95 Where $\varphi$ is used ¶ ↺
  - depends_on remark 7.94 Three numbers, two kinds ¶
- depends_on proposition 7.85 Irrationality of $\pi$ ¶
  - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
  - depends_on remark 7.87 Priority, and what irrationality does not give ¶
depends_on lemma 7.54 Truncation error ¶
- depends_on corollary 7.55 The decimal expansion ¶
- depends_on proposition 7.56 Irrationality ¶
  - depends_on remark 7.94 Three numbers, two kinds ¶ ↺
- depends_on remark 7.58 An identity, not a recipe ¶
  - depends_on remark 7.70 What the condition says ¶
depends_on lemma 7.139 A countable union of null sets is null ¶
- depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶
- depends_on proposition 7.141 The critical exponent, and the Hausdorff dimension ¶
  - depends_on proposition 7.143 The Hausdorff dimension never exceeds the box dimension ¶ ↺
- depends_on remark 7.94 Three numbers, two kinds ¶ ↺
depends_on lemma A.288 The iteration converges ¶
- depends_on corollary A.289 The zero set is a graph ¶
- depends_on lemma A.290 $h$ is Lipschitz ¶
  - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
depends_on proposition 7.48 Ratio test ¶
- depends_on definition 7.53 $\ee$ ¶
  - depends_on lemma A.72 Iterated integral inequality ¶
    - depends_on theorem A.74 Flow of a time-dependent vector field ¶
      - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
  - depends_on lemma A.306 The standard smooth bump ¶
    - depends_on lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶
  - depends_on proposition 7.56 Irrationality ¶ ↺
  - depends_on proposition 7.57 The limit definition ¶
    - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶
    - depends_on remark 7.58 An identity, not a recipe ¶ ↺
- depends_on definition 7.59 The real exponential ¶
  - depends_on lemma 7.60 Functional equation of the exponential ¶
    - depends_on proposition 7.68 The natural base ¶ ↺
    - depends_on proposition 7.61 The logarithm ¶
      - depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
      - depends_on definition 7.62 Real powers ¶
      - depends_on lemma 7.67 The limit over real exponents ¶ ↺
      - depends_on proposition 7.68 The natural base ¶ ↺
      - depends_on proposition 7.63 Laws of real powers ¶
    - depends_on proposition 7.63 Laws of real powers ¶ ↺
- depends_on definition 8.3 Exponential, sine, cosine ¶
  - depends_on lemma A.786 From the half-strip to the half plane ¶
    - depends_on corollary A.788 The two parameters are the same parameter ¶
      - depends_on theorem A.789 Kirchhoff's contraction coefficient ¶
    - depends_on lemma A.787 From the potential strip to the half plane ¶
      - depends_on corollary A.788 The two parameters are the same parameter ¶ ↺
  - depends_on lemma A.787 From the potential strip to the half plane ¶ ↺
  - depends_on lemma A.785 The hodograph image ¶
    - depends_on lemma A.786 From the half-strip to the half plane ¶ ↺
- depends_on lemma 7.72 Addition theorems ¶
  - depends_on proposition 7.79 The polygon recursion ¶
    - depends_on corollary 7.81 The decimal expansion of $\pi$ ¶
      - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶
    - depends_on proposition 7.83 Viète's product ¶
      - depends_on remark 7.84 One construction, two faces ¶
    - depends_on remark 7.82 What Archimedes proved ¶
    - depends_on remark 7.80 Why these are the polygon perimeters ¶
    - depends_on remark 7.84 One construction, two faces ¶ ↺
  - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
    - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
    - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on proposition 7.83 Viète's product ¶ ↺
    - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
    - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶ ↺
  - depends_on proposition 7.83 Viète's product ¶ ↺
  - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
  - depends_on remark 7.88 $\pi$ elsewhere in this treatise ¶ ↺
  - depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺
- depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
- depends_on theorem 7.51 Termwise differentiation ¶
  - depends_on definition 7.74 $\pi$ ¶
    - depends_on lemma 7.75 The first quadrant ¶
      - depends_on proposition 7.76 $\pi$ as the circle constant ¶
      - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
      - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
    - depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
    - depends_on proposition 7.79 The polygon recursion ¶ ↺
    - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
    - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
  - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
  - depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
    - depends_on lemma 7.75 The first quadrant ¶ ↺
    - depends_on lemma 7.78 Chord, arc, tangent ¶ ↺
    - depends_on lemma 9.136 Wallis integrals ¶
      - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
    - depends_on proposition 7.76 $\pi$ as the circle constant ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶ ↺
    - depends_on proposition 125.2 Crystallographic restriction ¶ ↺
    - depends_on proposition 9.141 First properties ¶
      - depends_on proposition 9.143 A cubic quadrature between two turning points ¶
      - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
    - depends_on remark 7.80 Why these are the polygon perimeters ¶ ↺
  - depends_on lemma 7.75 The first quadrant ¶ ↺
  - depends_on proposition 9.22 The exponential and its derivative ¶
    - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
      - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
      - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      - depends_on theorem 9.26 Structure of the solutions ¶
depends_on theorem 7.50 Power series; radius of convergence ¶
- depends_on definition 7.59 The real exponential ¶ ↺
- depends_on definition 8.3 Exponential, sine, cosine ¶ ↺
- depends_on theorem 7.51 Termwise differentiation ¶ ↺
depends_on theorem 8.20 Taylor expansion ¶
- depends_on lemma A.608 An entire function with a quadratic bound on its real part ¶
  - depends_on proposition A.609 The exponent is a quadratic polynomial ¶
    - depends_on proposition A.611 The constraint on $\Gamma$ ¶
      - depends_on proposition A.612 The dictionary between $A$ and $\Gamma$ ¶
- depends_on proposition 8.23 Residue at a simple pole ¶
- depends_on theorem A.598 Hudson ¶
- depends_on theorem 8.21 Laurent expansion ¶
  - depends_on definition 8.22 Isolated singularities; residue ¶
    - depends_on proposition 8.23 Residue at a simple pole ¶ ↺
    - depends_on proposition 17.96 Poles give asymptotics ¶
    - depends_on theorem 8.24 Residue theorem ¶
      - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on example 8.25 A real integral by residues ¶
      - depends_on proposition 17.96 Poles give asymptotics ¶ ↺
      - depends_on proposition 17.61 Mean-square response of a damped resonator ¶
      - depends_on theorem 31.68 Kutta–Joukowski ¶
      - depends_on theorem 9.26 Structure of the solutions ¶ ↺
  - depends_on theorem 8.24 Residue theorem ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Monotone convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:899
`depends_on`	→	Series	declared	parts/02-mathematical-methods/05-real-analysis.tex:899
`depends_on`	←	Terminating or repeating decimals are rational	declared	parts/02-mathematical-methods/05-real-analysis.tex:2545
`depends_on`	←	Truncation error	declared	parts/02-mathematical-methods/05-real-analysis.tex:1164
`depends_on`	←	A countable union of null sets is null	declared	parts/02-mathematical-methods/05-real-analysis.tex:4733
`depends_on`	←	The iteration converges	declared	appendices/A-long-proofs.tex:14481
`depends_on`	←	Ratio test	declared	parts/02-mathematical-methods/05-real-analysis.tex:931
`depends_on`	←	Power series; radius of convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:993
`depends_on`	←	Taylor expansion	declared	parts/02-mathematical-methods/06-complex-analysis.tex:505
`proves`	←	ch:05-real-analysis@proof-27	declared	parts/02-mathematical-methods/05-real-analysis.tex:902