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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.