List of Mathematical Formulas: “Cauchy Limits. Theory

List of basic formulas and concepts related to Cauchy limits:

  1. Definition of the Cauchy limit:
  • A sequence (a_n) converges to a limit (L) if for any (\epsilon > 0) there exists a natural number (N) such that for all (n > N) the inequality holds:
    [
    |a_n – L| < \epsilon
    ]
  1. Cauchy criterion for sequences:
  • A sequence (a_n) is convergent (i.e. has a limit) if and only if for any ( \epsilon > 0) there exists a natural number (N) such that for all (m, n > N) the following holds:
    [
    |a_n – a_m| < \epsilon
    ]
  1. Properties of limits:
  • If ( \lim_{n \to \infty} a_n = L ) and ( \lim_{n \to \infty} b_n = M ), then:
  • Sum: ( \lim_{n \to \infty} (a_n + b_n) = L + M )
  • Product: ( \lim_{n \to \infty} (a_n \cdot b_n) = L \cdot M )
  • Quotient: ( \lim_{n \to \infty} \frac{a_n}{b_n} = \frac{L}{M} ) (for ( M \neq 0 ))
  1. Function limits:
  • If a function ( f(x) ) has a limit ( L ) as ( x \to a ), then:
    [
    \lim_{x \to a} f(x) = L
    ]
  1. Theorem on the limit of a composite function:
  • If ( \lim_{n \to \infty} a_n = L ) and ( f ) is continuous at the point ( L ), then:
    [
    \lim_{n \to \infty} f(a_n) = f(L)
    ]

От Math

Добавить комментарий

Ваш адрес email не будет опубликован. Обязательные поля помечены *