In mathematics, the root test (sometimes called the Cauchy root test or Cauchy's radical test) is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity
lim sup n → ∞ | a n | n , {\displaystyle \limsup _{n\rightarrow \infty }{\sqrt[{n}]{|a_{n}|}},}
where a n {\displaystyle a_{n}} are the terms of the series, and states that the series converges absolutely if this quantity is less than one, but diverges if it is greater than one. The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821).
Root test explanation
For a series
∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}}
the root test uses the number
C = lim sup n → ∞ | a n | n , {\displaystyle C=\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|a_{n}|}},}
where "lim sup" denotes the limit superior, possibly +∞. Note that if
lim n → ∞ | a n | n , {\displaystyle \lim _{n\rightarrow \infty }{\sqrt[{n}]{|a_{n}|}},}
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. ∑ 1 / n 2 {\displaystyle \textstyle \sum 1/{n^{2}}} , and there are others for which C = 1 and the series diverges, e.g. ∑ 1 / n {\displaystyle \textstyle \sum 1/n} .
Application to power series This test can be used with a power series
f ( z ) = ∑ n = 0 ∞ c n ( z − p ) n {\displaystyle f(z)=\sum _{n=0}^{\infty }c_{n}(z-p)^{n}}
where the coefficients cn, and the center p are complex numbers and the argument z is a complex variable. The terms of this series would then be given by an = cn(z − p)n. One then applies the root test to the an as above. Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately). A corollary of the root test applied to a power series is the Cauchy–Hadamard theorem: the radius of convergence is exactly 1 / lim sup n → ∞ | c n | n , {\displaystyle 1/\limsup _{n\rightarrow \infty }{\sqrt[{n}]{|c_{n}|}},} taking care that we really mean ∞ if the denominator is 0.
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