Calc09_4 - 9.4 Radius of Convergence Greg Kelly Hanford...

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9.4 Radius of Convergence Greg Kelly, Hanford High School, Richland, Washington
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Convergence The series that are of the most interest to us are those that converge . Today we will consider the question: “Does this series converge, and if so, for what values of x does it converge?”
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The first requirement of convergence is that the terms must approach zero. n th term test for divergence 1 n n a = diverges if fails to exist or is not zero. lim n n a →∞ Note that this can prove that a series diverges , but can not prove that a series converges. Ex. 2: 0 ! n n n x = If then grows without bound. 1 x ! n n x If then 0 1 x < < 1 ! lim ! lim n n n n x n n x →∞ →∞ = = ∞ As , eventually is larger than , therefore the numerator grows faster than the denominator. n → ∞ n 1 x The series diverges. (except when x=0)
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(As in the previous example.) There are three possibilities for power series convergence. 1 The series converges over some finite interval: (the interval of convergence ). The series may or may not converge at the endpoints of the interval. There is a positive number R such that the series diverges for but converges for . x a R - x a R - < 2 The series converges for every x . ( ) R = ∞ 3 The series converges at and diverges everywhere else. ( ) 0 R = x a = The number R is the radius of convergence .
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