proposition 7.48 Ratio test

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:928 · p. 217

Rests on

depends_on proposition 7.47 Comparison; absolute convergence ¶
- 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 ¶
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 ¶

Supports

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.548 The joint flow is a translation action on the level set ¶
      - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
- 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 theorem A.311 General Stokes theorem ¶
- 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 7.70 What the condition says ¶
  - 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 remark 7.70 What the condition says ¶ ↺
    - depends_on definition 7.62 Real powers ¶
      - depends_on definition 7.90 The golden ratio ¶
      - depends_on proposition 7.63 Laws of real powers ¶
      - depends_on proposition 7.89 Irrationality of square roots ¶
    - 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 9.144 The pendulum equation at finite amplitude ¶
  - 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 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 ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Comparison; absolute convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:931
`depends_on`	→	Geometric series	declared	parts/02-mathematical-methods/05-real-analysis.tex:931
`depends_on`	→	Cauchy criterion	declared	parts/02-mathematical-methods/05-real-analysis.tex:931
`depends_on`	←	$\ee$	declared	parts/02-mathematical-methods/05-real-analysis.tex:1147
`depends_on`	←	The real exponential	declared	parts/02-mathematical-methods/05-real-analysis.tex:1328
`depends_on`	←	Exponential, sine, cosine	declared	parts/02-mathematical-methods/06-complex-analysis.tex:77
`depends_on`	←	Addition theorems	declared	parts/02-mathematical-methods/05-real-analysis.tex:1834
`depends_on`	←	Irrationality of $\pi$	declared	parts/02-mathematical-methods/05-real-analysis.tex:2472
`depends_on`	←	Termwise differentiation	declared	parts/02-mathematical-methods/05-real-analysis.tex:1018
`proves`	←	ch:05-real-analysis@proof-29	declared	parts/02-mathematical-methods/05-real-analysis.tex:934