The D’Alembert criterion is used to determine the convergence of series. Here are the main formulas and conditions associated with this criterion:

  1. Definition: For a series (\sum a_n) (where (a_n) are the terms of the series), the d’Alembert criterion states that if there exists a limit:

[
L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|
]

that:

  • If (L < 1), the series (\sum a_n) converges.
  • If (L > 1) or (L = \infty), the series (\sum a_n) diverges.
  • If (L = 1), the feature does not provide information about the convergence of the series.
  1. Application: To apply the feature, it is necessary to calculate the limit (\frac{a_{n+1}}{a_n}) and analyze its value.
  2. Example: Consider the series (\sum \frac{1}{n!}):
  • (a_n = \frac{1}{n!})
  • We calculate (\frac{a_{n+1}}{a_n} = \frac{1/(n+1)!}{1/n!} = \frac{1}{n+1})
  • We find the limit: (L = \lim_{n \to \infty} \frac{1}{n+1} = 0 < 1), therefore, the series converges.

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