|
In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity : 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. == 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. . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Root test」の詳細全文を読む スポンサード リンク
|