where 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
Decision diagram for the root test
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
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 such a power series is the Cauchy–Hadamard theorem: the radius of convergence is exactly taking care that we really mean ∞ if the denominator is 0.
The proof of the convergence of a series Σan is an application of the comparison test. If for all n ≥ N (N some fixed natural number) we have then . Since the geometric series converges so does by the comparison test. Hence Σan converges absolutely.
If for infinitely many n, then an fails to converge to 0, hence the series is divergent.
Proof of corollary:
For a power series Σan = Σcn(z − p)n, we see by the above that the series converges if there exists an N such that for all n ≥ N we have
for all n ≥ N, which implies that in order for the series to converge we must have for all sufficiently large n. This is equivalent to saying
so 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
^Bottazzini, Umberto (1986), The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag, pp. 116–117, ISBN 978-0-387-96302-0. Translated from the Italian by Warren Van Egmond.
^Briggs, William; Cochrane, Lyle (2011). Calculus: Early Transcendentals. Addison Wesley. p. 571.
Knopp, Konrad (1956). "§ 3.2". Infinite Sequences and Series. Dover publications, Inc., New York. ISBN 0-486-60153-6.
Whittaker, E. T. & Watson, G. N. (1963). "§ 2.35". A Course in Modern Analysis (fourth ed.). Cambridge University Press. ISBN 0-521-58807-3.
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