Editing Geometric series (section)

== Connection to the power series ==
Like the geometric series, a [[power series]] has one parameter for a common variable raised to successive powers corresponding to the geometric series's <math> r </math>, but it has additional parameters <math>a_0, a_1, a_2, \ldots,</math> one for each term in the series, for the distinct coefficients of each <math>x^0, x^1, x^2, \ldots</math>, rather than just a single additional parameter <math>a</math> for all terms, the common coefficient of <math>r^k</math> in each term of a geometric series. The geometric series can therefore be considered a class of power series in which the sequence of coefficients satisfies <math>a_k = a</math> for all <math>k</math> and <math>x = r</math>.{{sfnp|Apostol|1967|pp=389}}

This special class of power series plays an important role in mathematics, for instance for the study of [[ordinary generating functions]] in combinatorics and the [[Summation method|summation]] of divergent series in analysis. Many other power series can be written as transformations and combinations of geometric series, making the geometric series formula a convenient tool for calculating formulas for those power series as well.{{r|wilf|bo}}

As a power series, the geometric series has a [[radius of convergence]] of 1.{{r|spivak}} This could be seen as a consequence of the [[Cauchy–Hadamard theorem]] and the fact that <math display="block">\lim_{n \rightarrow \infty}\sqrt[n]{a} = 1</math> for any <math>a</math> or as a consequence of the [[ratio test]] for the convergence of infinite series, with <math display="block">\lim_{n \rightarrow \infty} \frac{|a r^{n+1}| }{ |a r^{n}|} = |r|</math>  implying convergence only for <math>|r| < 1.</math> However, both the ratio test and the Cauchy–Hadamard theorem are proven using the geometric series formula as a logically prior result, so such reasoning would be subtly circular.{{sfnp|Spivak|2008|p=476}}