Root test
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity where formula_2 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. It is particularly useful in connection with power series. Root test explanation. The root test was developed first by Augustin-Louis Cauchy who published it in his textbook Cours d'analyse (1821). Thus, it 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: There are some series for which "C" = 1 and the series converges, e.g. formula_6, and there are others for which "C" = 1 and the series diverges, e.g. formula_7. Application to power series. This test can be used with a power series where the coefficients "c", 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 "a" = "c"("z" − "p"). One then applies the root test to the "a" 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 such a  power series is the Cauchy–Hadamard theorem: the radius of convergence is exactly formula_9 taking care that we really mean ∞ if the denominator is 0. Proof. The proof of the convergence of a series Σ"a" is an application of the comparison test. If for all "n" ≥ "N" ("N" some fixed natural number) we have formula_10, then formula_11. Since the geometric series formula_12 converges so does formula_13 by the comparison test. Hence Σ"a" converges absolutely. If formula_14 for infinitely many "n", then "a" fails to converge to 0, hence the series is divergent. Proof of corollary: For a power  series Σ"a" = Σ"c"("z" − "p"), we see by the above that the series converges if there exists an "N" such that for all "n" ≥ "N" we have equivalent to for all "n" ≥ "N", which implies that in order for the series to converge we must have formula_17 for all sufficiently large "n". This is equivalent to saying so formula_19 Now the only other place where convergence is possible is when (since points > 1 will diverge) and this will not change the radius of convergence since these are just the points lying on the boundary of the  interval or disc, so Examples. "Example 1:" Applying the root test and using the fact that formula_23 Since formula_25 the series diverges. "Example 2:" The root test shows convergence because This example shows how the root test is stronger than the ratio test. The ratio test is inconclusive for this series if formula_28 is odd so formula_29 (though not if formula_28 is even), because Root tests hierarchy. Root tests hierarchy is built similarly to the ratio tests hierarchy (see Section 4.1 of ratio test, and more specifically Subsection 4.1.4 there). For a series formula_32 with positive terms we have the  following tests for convergence/divergence. Let formula_33 be an integer, and let formula_34 denote the formula_35th iterate of natural logarithm, i.e. formula_36 and for any formula_37, formula_38. Suppose that formula_39, when formula_28 is large, can be presented in the form Proof. Since formula_44, then we have From this, From Taylor's expansion applied to the right-hand side, we obtain: Hence, The final result follows from the integral test for convergence.