Mathematical Analysis

Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley.

Type

Book

Intellectual base

Textbook

Year

1974

Pages cited

6-7

ISBN

9780201002881

Link

https://archive.org/details/mathematicalanal0000apos

Cited from

mathematics

Cited by

8 citations across 8 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Boundedness
  • Boundedness is not the same as convergence, as Apostol (1974) develops in his treatment of bounded sequences and the Bolzano-Weierstrass theorem.

    This sourcedevelops the Bolzano-Weierstrass theorem (every bounded sequence in R^n has a convergent subsequence), emphasizing boundedness as the essential hypothesis distinguishing it from convergence.

Domain-specific

• Conditional convergence
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Decimal representation
  • … fractional sequence, weighted power-of-ten sum and convergence, terminating convention, dual trailing-nine representation, sign and locale convention and rational periodicity criterion are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Finite set
  • … bijection to an initial finite ordinal, cardinality including zero, equivalent induction and Dedekind criteria, subsets, unions and products, and contrast with infinite sets are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Improper integral
  • … truncation parameters, proper subintegrals, one-sided independent limits, finite convergence criterion, absolute or conditional status and distinction from Cauchy principal value are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Real-valued function
  • A real-valued function assigns exactly one real number to each element of its declared domain, which may itself be arbitrary rather than real.
• Riemann sum
  • For a tagged partition \(P\), a Riemann sum is \(S(f,P)=\sum_{i=1}^n f(\xi_i)\Delta x_i\); Riemann integrability means these sums approach one common limit as the mesh tends to zero uniformly over admissible tag choices.
• Root Test
  • It does not prove divergence merely because \(L=1\): both \(\sum1/n\) and \(\sum1/n^2\) have root-test value one.

Verification

Does it exist? Confirmed. This work's ISBN resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Not recorded. Neither this nor any other of the 8 citations of this work carries a recorded support check.

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