proposition 7.47 Comparison; absolute convergence

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:910 · 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 ¶ ↺
depends_on theorem 7.8 Cauchy criterion ¶
- depends_on corollary A.47 Cauchy completeness ¶
  - depends_on corollary A.46 Monotone convergence ¶ ↺
  - depends_on theorem A.40 Least-upper-bound property ¶ ↺
  - proves proof app:A-long-proofs@proof-34 ¶
- depends_on definition 7.4 Convergence ¶ ↺
- proves proof ch:05-real-analysis@proof-5 ¶
proves proof ch:05-real-analysis@proof-28 ¶

Supports

depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
- depends_on proposition 9.139 Series for the complete integral of the first kind ¶
  - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
- 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 corollary 9.27 Linear stability ¶
      - depends_on proposition 9.29 The planar classification ¶
    - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶
    - 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 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on theorem 9.26 Structure of the solutions ¶
      - depends_on corollary 9.27 Linear stability ¶ ↺
- depends_on theorem 9.8 Picard–Lindelöf ¶
  - depends_on corollary 32.5 Trajectories do not cross ¶
    - depends_on definition 32.29 Limit cycle ¶
      - depends_on phenomenon 32.35 Self-sustained oscillation ¶
    - depends_on definition 32.38 Poincaré section and return map ¶
      - depends_on definition 32.91 The circle map ¶
      - depends_on example 32.70 The Hénon map ¶
      - depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶
      - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
      - depends_on proposition 32.39 Stability of a periodic orbit ¶
    - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
      - depends_on corollary 32.32 No chaos in a planar flow ¶
  - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
    - depends_on definition 9.37 Fundamental matrix ¶
      - depends_on proposition 9.38 Liouville's formula ¶
      - depends_on theorem 9.39 Floquet ¶
    - depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
      - depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
      - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
    - depends_on proposition 20.9 Linear resistance: exact motion, and the lost range ¶
      - depends_on remark 20.11 What has no closed form, and what to do about it ¶
    - depends_on proposition 9.103 The positive zeros and their spacing ¶
    - depends_on proposition 9.18 The linear equation: integrating factor ¶
      - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶ ↺
    - 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 theorem 9.23 Solution of a constant-coefficient system ¶ ↺
    - depends_on theorem 9.13 Dimension of the solution space ¶
      - depends_on definition 9.37 Fundamental matrix ¶ ↺
      - depends_on proposition 28.18 The three regimes ¶
      - depends_on proposition 28.4 General solution and isochronism ¶
      - depends_on proposition 9.106 The modified series and what it does not do ¶
      - depends_on proposition 9.28 The scalar constant-coefficient equation ¶
      - depends_on proposition 9.16 Wronskian test for independence ¶
  - depends_on proposition 32.4 The flow is a one-parameter group ¶
    - depends_on corollary 32.5 Trajectories do not cross ¶ ↺
  - depends_on proposition 10.16 Cauchy's characteristic strips ¶
  - depends_on proposition 20.7 Terminal speed and the approach to it ¶
    - depends_on example 20.8 A raindrop ¶
  - depends_on proposition 9.11 Continuous dependence on the initial data ¶
  - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
  - 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 theorem 9.34 Poincaré–Bendixson; quoted ¶
    - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶ ↺
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 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 remark 7.94 Three numbers, two kinds ¶
  - 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 remark 7.70 What the condition says ¶
- 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 remark 7.94 Three numbers, two kinds ¶ ↺
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - 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 125.3 What the theorem does and does not forbid ¶
    - 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 proposition 8.4 Euler's formula ¶
- depends_on lemma A.786 From the half-strip to the half plane ¶ ↺
- depends_on lemma 8.11 The fundamental $2\pi\ii$ ¶
  - 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 8.16 Cauchy integral formula ¶
    - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on theorem 8.18 Liouville ¶
    - depends_on theorem 8.21 Laurent expansion ¶
      - depends_on definition 8.22 Isolated singularities; residue ¶ ↺
      - depends_on theorem 8.24 Residue theorem ¶ ↺
  - depends_on theorem 8.21 Laurent expansion ¶ ↺
  - depends_on theorem 8.24 Residue theorem ¶ ↺
- depends_on lemma 17.14 Uniformly bounded sine sums ¶
  - depends_on lemma 17.15 A polynomial whose partial sum spikes at the origin ¶
    - depends_on proposition 17.16 An explicit continuous function with a divergent Fourier series ¶
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 7.51 Termwise differentiation ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Monotone convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:914
`depends_on`	→	Series	declared	parts/02-mathematical-methods/05-real-analysis.tex:914
`depends_on`	→	Cauchy criterion	declared	parts/02-mathematical-methods/05-real-analysis.tex:914
`depends_on`	←	Weierstrass $M$-test; uniform limits are continuous	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:207
`depends_on`	←	Ratio test	declared	parts/02-mathematical-methods/05-real-analysis.tex:931
`depends_on`	←	Euler's formula	declared	parts/02-mathematical-methods/06-complex-analysis.tex:87
`depends_on`	←	Power series; radius of convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:993
`depends_on`	←	Termwise differentiation	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`proves`	←	ch:05-real-analysis@proof-28	declared	parts/02-mathematical-methods/05-real-analysis.tex:917