definition 7.45 Series

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:889 · p. 217

ground object -- no derivation owed

Rests on

depends_on definition 7.2 Absolute value ¶
depends_on definition 7.4 Convergence ¶
- depends_on definition 7.2 Absolute value ¶ ↺

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 11.4 Expectation ¶
- depends_on definition A.741 The statistical setting ¶
- depends_on definition 11.23 Expectation, moments and quantiles in the continuous case ¶
  - depends_on example 11.30 Exponential lifetimes ¶
  - depends_on proposition 11.29 Gaussian moments ¶
  - depends_on proposition 11.27 Probability integral transform ¶
  - depends_on proposition 11.24 The algebra of the discrete case transfers ¶
    - depends_on example 11.30 Exponential lifetimes ¶ ↺
    - depends_on proposition 11.26 Product rule ¶
- depends_on proposition 11.5 Transfer, linearity, monotonicity ¶
  - depends_on definition 11.11 Covariance and correlation ¶
    - depends_on corollary 11.15 Bounds on the correlation coefficient ¶
    - depends_on definition 11.16 Sample mean, variance and correlation ¶
      - depends_on example 11.19 A counting measurement ¶
      - depends_on example 11.31 A length measurement ¶
      - depends_on proposition 11.17 Sampling identities ¶
    - depends_on lemma 11.63 Cauchy–Schwarz for random variables ¶
      - depends_on corollary 11.15 Bounds on the correlation coefficient ¶ ↺
      - depends_on corollary 11.65 When the bound is attained ¶
      - depends_on lemma A.199 The standardised third moment is at least one ¶
      - depends_on theorem 11.64 Cramér–Rao inequality ¶
    - depends_on proposition 11.12 Bilinearity ¶
      - depends_on example 11.18 Three standard discrete distributions ¶
      - depends_on proposition 11.14 Independence implies zero covariance; the converse fails ¶
      - depends_on proposition 11.17 Sampling identities ¶ ↺
  - depends_on definition 11.6 Moments, variance, standard deviation ¶
    - depends_on definition 11.11 Covariance and correlation ¶ ↺
    - depends_on definition 11.16 Sample mean, variance and correlation ¶ ↺
    - depends_on example 11.18 Three standard discrete distributions ¶ ↺
    - depends_on lemma 11.63 Cauchy–Schwarz for random variables ¶ ↺
    - depends_on lemma 11.50 Markov and Chebyshev inequalities ¶
      - depends_on corollary 11.51 Weak law of large numbers ¶
      - depends_on theorem 11.101 Davies' bound ¶
    - depends_on proposition 11.7 Variance identity ¶
      - depends_on proposition 11.10 What the mean and the median minimise ¶
  - depends_on lemma 11.50 Markov and Chebyshev inequalities ¶ ↺
  - depends_on proposition 11.12 Bilinearity ¶ ↺
  - depends_on proposition 11.24 The algebra of the discrete case transfers ¶ ↺
  - depends_on proposition 11.7 Variance identity ¶ ↺
- depends_on theorem A.169 Dominated convergence ¶
- depends_on theorem A.170 Fubini's theorem, bounded case ¶
- depends_on theorem A.223 Tonelli and dominated convergence, as used here ¶
depends_on definition 11.1 Discrete probability space ¶
- depends_on definition 11.3 Discrete random variable ¶
  - depends_on definition 11.4 Expectation ¶ ↺
  - depends_on definition 11.13 Independence ¶
    - depends_on definition 11.25 Joint density and independence ¶
      - depends_on proposition 11.26 Product rule ¶ ↺
    - depends_on definition 11.16 Sample mean, variance and correlation ¶ ↺
    - depends_on proposition 11.14 Independence implies zero covariance; the converse fails ¶ ↺
  - depends_on definition 11.8 Quantiles and median ¶
    - depends_on example 11.31 A length measurement ¶ ↺
    - depends_on proposition 11.10 What the mean and the median minimise ¶ ↺
- depends_on definition 11.34 Measure; measure space ¶
  - depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶
    - depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶
  - depends_on definition 11.35 Null set; almost everywhere ¶
    - depends_on definition 11.38 Measure-preserving map; invariant measure; ergodicity ¶ ↺
    - depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶
      - depends_on theorem 11.39 Birkhoff's pointwise ergodic theorem ¶ ↺
  - depends_on proposition 11.36 Monotonicity, countable subadditivity, and null sets ¶ ↺
- depends_on definition 11.33 Sigma-algebra ¶
  - depends_on definition 11.34 Measure; measure space ¶ ↺
- depends_on proposition 11.2 Elementary rules ¶
  - depends_on proposition 11.5 Transfer, linearity, monotonicity ¶ ↺
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 proposition 7.52 Alternating series test ¶
- depends_on lemma 7.73 $\cos 2 < 0$ ¶
  - 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 corollary 7.81 The decimal expansion of $\pi$ ¶
      - depends_on proposition 7.83 Viète's product ¶
      - 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 proposition 125.2 Crystallographic restriction ¶ ↺
depends_on proposition 7.49 Cauchy product ¶
- depends_on lemma 7.60 Functional equation of the exponential ¶ ↺
- 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 proposition 7.79 The polygon recursion ¶ ↺
  - 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 7.72 Addition theorems ¶
  - depends_on proposition 7.79 The polygon recursion ¶ ↺
  - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
  - 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 7.47 Comparison; absolute convergence ¶
- 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 9.22 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 9.8 Picard–Lindelöf ¶
    - depends_on corollary 32.5 Trajectories do not cross ¶
      - depends_on definition 32.29 Limit cycle ¶
      - depends_on definition 32.38 Poincaré section and return map ¶
      - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
    - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on definition 9.37 Fundamental matrix ¶
      - depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
      - depends_on proposition 20.9 Linear resistance: exact motion, and the lost range ¶
      - depends_on proposition 9.103 The positive zeros and their spacing ¶
      - depends_on proposition 9.18 The linear equation: integrating factor ¶
      - depends_on theorem A.74 Flow of a time-dependent vector field ¶ ↺
      - depends_on theorem 9.23 Solution of a constant-coefficient system ¶ ↺
      - depends_on theorem 9.13 Dimension of the solution space ¶
    - 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 definition 7.59 The real exponential ¶ ↺
  - 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 lemma A.787 From the potential strip to the half plane ¶
    - 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 7.85 Irrationality of $\pi$ ¶ ↺
  - depends_on theorem 7.51 Termwise differentiation ¶
    - depends_on definition 7.74 $\pi$ ¶ ↺
    - 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 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 theorem 8.16 Cauchy integral formula ¶
      - depends_on theorem 8.17 Derivatives of all orders; Cauchy estimates ¶
      - depends_on theorem 8.21 Laurent expansion ¶
    - 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 ¶ ↺
depends_on proposition 7.46 Geometric series ¶
- depends_on lemma 7.86 Terminating or repeating decimals are rational ¶ ↺
- depends_on lemma 7.54 Truncation error ¶
  - depends_on corollary 7.55 The decimal expansion ¶
  - depends_on proposition 7.56 Irrationality ¶ ↺
  - depends_on remark 7.58 An identity, not a recipe ¶ ↺
- 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 theorem 7.50 Power series; radius of convergence ¶ ↺
- 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 8.23 Residue at a simple pole ¶ ↺
  - depends_on theorem A.598 Hudson ¶
  - depends_on theorem 8.21 Laurent expansion ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Absolute value	declared	parts/02-mathematical-methods/05-real-analysis.tex:893
`depends_on`	→	Convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:893
`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`	←	Expectation	declared	parts/02-mathematical-methods/09-probability-statistics.tex:142
`depends_on`	←	Discrete probability space	declared	parts/02-mathematical-methods/09-probability-statistics.tex:79
`depends_on`	←	Terminating or repeating decimals are rational	declared	parts/02-mathematical-methods/05-real-analysis.tex:2545
`depends_on`	←	Alternating series test	declared	parts/02-mathematical-methods/05-real-analysis.tex:1109
`depends_on`	←	Cauchy product	declared	parts/02-mathematical-methods/05-real-analysis.tex:953
`depends_on`	←	Comparison; absolute convergence	declared	parts/02-mathematical-methods/05-real-analysis.tex:914
`depends_on`	←	Geometric series	declared	parts/02-mathematical-methods/05-real-analysis.tex:899