Ratio Test/Warning

Ratio Test: Warning

Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series of real numbers in $\R$, or a series of complex numbers in $\C$.

Let the sequence $\sequence {a_n}$ satisfy:

$\ds \lim_{n \mathop \to \infty} \size {\frac {a_{n + 1} } {a_n} } = l$


If $l = 1$, the ratio test provides no information on whether $\ds \sum_{n \mathop = 1}^\infty a_n$ converges absolutely, converges conditionally, or diverges.

If $\size {\dfrac {a_{n + 1} } {a_n} } \to \infty$ as $n \to \infty$, then of course $\ds \sum_{n \mathop = 1}^\infty a_n$ diverges.


Sources

  • 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 6.17$
  • 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.2$: Summary of convergence tests: Theorem $1.6$
  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ratio test
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ratio test
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): ratio test
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