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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
proof of radius of convergence
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(Proof)
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"proof of radius of convergence" is owned by mathwizard.
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(view preamble)
Cross-references: ratio test, radius of convergence, right hand side, divergent, series, absolutely convergent, power series, Cauchy's root test
This is version 3 of proof of radius of convergence, born on 2003-01-10, modified 2005-07-04.
Object id is 3890, canonical name is ProofOfRadiusOfConvergence.
Accessed 4075 times total.
Classification:
| AMS MSC: | 30B10 (Functions of a complex variable :: Series expansions :: Power series ) | | | 40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions) |
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Pending Errata and Addenda
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![$\displaystyle \limsup_{k\to\infty}\sqrt[k]{\vert a_k(x-x_0)^k\vert}=\vert x-x_0\vert\limsup_{k\to\infty}\sqrt[k]{\vert a_k\vert}<1.$](http://images.planetmath.org:8080/cache/objects/3890/l2h/img1.png "$\displaystyle \limsup_{k\to\infty}\sqrt[k]{\vert a_k(x-x_0)^k\vert}=\vert x-x_0\vert\limsup_{k\to\infty}\sqrt[k]{\vert a_k\vert}<1.$"){kind=small}
![$\displaystyle \vert x-x_0\vert<\frac{1}{\limsup_{k\to\infty}\sqrt[k]{\vert a_k\vert}}=\liminf_{k\to\infty}\frac{1}{\sqrt[k]{\vert a_k\vert}}=.$](http://images.planetmath.org:8080/cache/objects/3890/l2h/img2.png "$\displaystyle \vert x-x_0\vert<\frac{1}{\limsup_{k\to\infty}\sqrt[k]{\vert a_k\vert}}=\liminf_{k\to\infty}\frac{1}{\sqrt[k]{\vert a_k\vert}}=.$")
![$\displaystyle \vert x-x_0\vert>\liminf_{k\to\infty}\frac{1}{\sqrt[k]{\vert a_k\vert}},$](http://images.planetmath.org:8080/cache/objects/3890/l2h/img3.png "$\displaystyle \vert x-x_0\vert>\liminf_{k\to\infty}\frac{1}{\sqrt[k]{\vert a_k\vert}},$")
