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Convergence
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An important concept used in working with series in complex analysis is the radius of convergence R Simply put, we want to know over what region R of the complex plane does the series converge It may be that the series converges everywhere, or it could turn out that the series only converges inside the unit disc, say
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Sequences and Series
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First let s take a look at sequences again If each term in a sequence is larger than or equal to the previous term, which means that an+1 an We say that the sequence is monotonic increasing On the other hand, if an+1 an then the sequence is monotonic decreasing If each term in the sequence is bounded above by some constant M: an < M (59)
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then we say that the sequence is bounded A bounded monotonic sequence (either increasing or decreasing) converges
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CAUCHY S CONVERGENCE CRITERION
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Saying that a sequence converges is the same as saying that it has a limit, so we can formalize the notion of convergence Leave it up to Cauchy to have done that for us So, {an } converges if given an > 0 we can nd an N such that am an < for m, n > N
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Cauchy s convergence criterion is necessary and suf cient to show convergence of a sequence
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CONVERGENCE OF A COMPLEX SERIES
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Remember that any complex function f ( z ) can be written in terms of real and imaginary parts, just like a complex number The real and imaginary parts are themselves real functions So one way to check convergence is to check the convergence of the real and imaginary parts assuming we have a series representation available and seeing if they converge So a necessary and suf cient condition that a series of the form a j + ib j converges is that the two series a j and b j both converge
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Complex Variables Demysti ed
Convergence Tests
The following convergence tests can be used to evaluate whether or not a series converges If we say that a series an converges absolutely, we mean that
(510)
converges The rst test that we can apply for convergence is the comparison test The comparison test tells us that if bn converges and an bn , then the series an converges absolutely If bn diverges and an bn , the series an also diverges However, we can t say anything about the series an The ratio test is a nice test that appeals to common sense We take the ratio of the terms an+1 to an and take the limit n Let an+1 =R an
(511)
There are two possibilities: If R < 1 then the series converges absolutely If R > 1 then the series is divergent If R = 1then no information is available from the test The nth root test checks the limit: lim n an = R
(512)
The possibilities here are the same we encountered with the ratio test These are If R < 1 then the series converges absolutely If R > 1 then the series is divergent If R = 1then no information is available from the test
Sequences and Series
Raabe s test checks the limit: a lim n 1 n+1 n an Again If R < 1 then the series converges absolutely If R > 1 then the series is divergent If R = 1then no information is available from the test Finally, we consider the Weierstrass M-test Suppose that an ( z ) M n If M n does not depend on z in some region of the complex plane where an ( z ) M n holds and M n converges, then an ( z ) is uniformly convergent (513)
Uniformly Converging Series
We often nd in the limits we compute that N depends on When a series is uniformly convergent, then for any > 0 there is an N not depending on such that an ( z ) R < for n > N , where R is the limit That is, if the same N holds for all z in a given region D of the complex plane, then we say that the convergence is uniform