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 0≤x<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:
Answer
The convergence is absolute, provided that 0≤x<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 an→0, which means that |an|≤C and so ∑n≥0|anxn|≤C∑n≥0|x|n<∞ for 0≤x<1.
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