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Math and science::Analysis::Tao::07. Series

Ratio test

Let \( \sum_{n=m}^{\infty}a_n \) be a series of non-zero numbers.

  • If [...], then the series \( \sum_{n=m}^{\infty}a_n \) is absolutely convergent (hence conditionally convergent).
  • If [...], then the series \( \sum_{n=m}^{\infty}a_n \) is not conditionally convergent (hence not absolutely convergent).
  • Otherwise, we cannot assert any conclusion.

The non-zero requirement is needed to avoid a division by zero.