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Notes on Absolute Convergence 1

# Notes on Absolute Convergence 1 - Notes on Absolute...

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Notes on Absolute Convergence So far the only series that convergence tests apply are either series of positive terms or alternating series. There are series that do not fit either category. The notion of absolute convergence is partial designed to address this issue. Definition : Let 0 n n a = denote a series of numbers. The series 0 n n a = is said to converge absolutely (or is absolutely convergent ) provided the series 0 n n a = converges. Theorem : If a series converse absolutely, then it converges. Observations : There are several observations that one needs to keep in mind when applying this theorem. In showing a series 0 n n a = converges absolutely, one does not apply a test specifically to the series 0 n n a = , but rather one creates the series of absolute values 0 n n a = and applies the appropriate test to 0 n n a = . The series 0 n n a = is a series of positive terms. So only those convergence tests that apply to series of positive terms can be used to show it converges, and hence that the original series 0 n n

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