Absolute Convergence

Absolute Convergence - Absolute Convergence When we first...

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When we first talked about series convergence we briefly mentioned a stronger type of convergence but didn’t do anything with it because we didn’t have any tools at our disposal that we could use to work problems involving it. We now have some of those tools so it’s now time to talk about absolute convergence in detail. First, let’s go back over the definition of absolute convergence. Definition A series is called absolutely convergent if is convergent. If is convergent and is divergent we call the series conditionally convergent . We also have the following fact about absolute convergence. Fact If is absolutely convergent then it is also convergent. Proof First notice that is either or it is depending on its sign. This means that we can then say, Now, since we are assuming that is convergent then is also convergent since we can just factor the 2 out of the series and 2 times a finite value will still be finite. This however allows us to use the Comparison Test to say that is also a convergent series. Finally, we can write,
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This note was uploaded on 11/10/2011 for the course MATH 136 taught by Professor Prellis during the Fall '08 term at Rutgers.

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Absolute Convergence - Absolute Convergence When we first...

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