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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 \( \limsup_{n \rightarrow \infty} \frac{|a_{n+1}| }{|a_n|} < 1 \), then the series \( \sum_{n=m}^{\infty}a_n \) is absolutely convergent (hence conditionally convergent).
  • If \( \limsup_{n \rightarrow \infty} \frac{|a_{n+1}| }{|a_n|} > 1 \), 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.


Intuition

Tao proves this result using the Root Test along with a lemma. However, for understanding and remembering the Ratio Test, I think it is best described in terms of the behaviour of the geometric series.

For a geometric series, \( \sum_{n=m}^{\infty} a_n = \sum_{n=m}^{\infty} r^{n} = r^m + r^{m+1} + r^{m+2} ... \), we know that it converges absolutely if \( r < 1 \) and diverges otherwise.

This criteria can be rephrased using the ratio between the terms: \( \frac{a_{n+1}}{a_n} = r < 1 \) implies that the series converges (and diverges otherwise). If we extend this intuition to all series, we have the ratio test.