Root Test
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity limsup_{nrightarrowinfty}sqrt[n]{ a_n }, where an are the terms of the series, and states thats the series converges absolutely if this quantity is less than 1 but diverges if it is greater than 1. It is particularly useful in connection with power series. The terms of this series would then be given by an = cn(z p)n. One then applies...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the root test is a criterion for the convergence (a convergence test) of an infinite series. It depends on the quantity limsup_{nrightarrowinfty}sqrt[n]{ a_n }, where an are the terms of the series, and states thats the series converges absolutely if this quantity is less than 1 but diverges if it is greater than 1. It is particularly useful in connection with power series. 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 that the radius of convergence is exactly 1/limsup_{n rightarrow infty}{sqrt[n]{ c_n }}, taking care that we really mean if the denominator is 0.
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