Application To Power Series
This test can be used with a power 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 that the radius of convergence is exactly taking care that we really mean ∞ if the denominator is 0.
Read more about this topic: Root Test
Famous quotes containing the words application to, application, power and/or series:
“Five oclock tea is a phrase our rude forefathers, even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for all the ills that flesh is heir to, the glorious Magna Charta.”
—Lewis Carroll [Charles Lutwidge Dodgson] (18321898)
“The receipt to make a speaker, and an applauded one too, is short and easy.Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (16941773)
“The delicious faces of children, the beauty of school-girls, the sweet seriousness of sixteen, the lofty air of well-born, well-bred boys, the passionate histories in the looks and manners of youth and early manhood, and the varied power in all that well-known company that escort us through life,we know how these forms thrill, paralyze, provoke, inspire, and enlarge us.”
—Ralph Waldo Emerson (18031882)
“The womans world ... is shown as a series of limited spaces, with the woman struggling to get free of them. The struggle is what the film is about; what is struggled against is the limited space itself. Consequently, to make its point, the film has to deny itself and suggest it was the struggle that was wrong, not the space.”
—Jeanine Basinger (b. 1936)