Friday, 6 November 2015

real analysis - Why is the convergence absolute?



There is one thing my book uses in a proof after Abels theorem which I do not understand:



Lets say that n=0an converges.




For 0x<1, we look at n=0anxn. The book says that this series converges absolutely for all the values of x we have defined it. But why? We started with a sequence that might not even converge absolutely. And how do we know that we even have convergence when we introduce the x variable? It would have been easy to see convergence if the orginal series was absolutely convergent, not only convergent, but they only state that the original series is convergent.



UPDATE:



If you are interested I got the picture from the book. Theorem 8.2 is Abels theorem, Definition 3.48 is the Cauchy-product(but this is clear from the picture), and what Theorem 3.51 states is also clear from the picture:



enter image description here


Answer



The convergence is absolute, provided that 0x<1. In other words, x=1 is not allowed (and Abel's theorem attempts to assign a sensible value to the sum at x=1). Since n=0an converges, then an0, which means that |an|C and so n0|anxn|Cn0|x|n< for 0x<1.



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