The Test
The root test was developed first by Augustin-Louis Cauchy and so is sometimes known as the Cauchy root test or Cauchy's radical test. For a series
the root test uses the number
where "lim sup" denotes the limit superior, possibly ∞. Note that if
converges then it equals C and may be used in the root test instead.
The root test states that:
- if C < 1 then the series converges absolutely,
- if C > 1 then the series diverges,
- if C = 1 and the limit approaches strictly from above then the series diverges,
- otherwise the test is inconclusive (the series may diverge, converge absolutely or converge conditionally).
There are some series for which C = 1 and the series converges, e.g., and there are others for which C = 1 and the series diverges, e.g. .
Read more about this topic: Root Test
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