Talk:Abel's Theorem

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Quick question: why remove the category "series" from the category list?

(You may equally ask me the same question: "why remove it from Calculus?" to which my answer would be "it's pre-calculus, it's a result that's relevant to series rather than integration directly.") --prime mover (talk) 23:15, 26 December 2008 (UTC)

I've got this defined as:

If $\displaystyle \sum_{k=0}^\infty a_k$ converges, then $\displaystyle \lim_{x \to 1^-} \left({\sum_{k=0}^\infty a_k x^k}\right) = \sum_{k=0}^\infty a_k$.

Anyone mind if I add this to the page, or use it to replace what's already there? --prime mover (talk) 23:23, 23 January 2009 (UTC)

Okay, I done what I threatened to do. The existing result is still there, commented out, in case anyone wants to restore it and throw away what I done. But I note that wikipedia [1] seems to confirm my own belief of what Abel's Theorem is. --prime mover (talk) 11:43, 25 January 2009 (UTC)

Joe was the one who originally put this in the wanted proofs list, so we should see what he thinks the theorem is. We can always put the other one on another page. --Cynic (talk) 16:57, 25 January 2009 (UTC)

I'm not at home at the moment, so I don't have my textbooks or notes with me. If I remember correctly, however, this theorem we have up is the real case of the more general Abel's Theorem in the complex plane, which, and again, I don't have my notes or anything, this is pure memory here, that if $\Sigma a_k \ $ is convergent, then $\lim \Sigma a_k x^k = \Sigma a_k \ $ where $x \to 1$ from inside the unit circle, between any two lines which intersect at 1. That's my memory... Zelmerszoetrop 19:05, 25 January 2009 (UTC)

Excellent, I thought as much. I'm going to address complex extensions to various of these real analysis theorems once I've done the appropriate groundwork there. --prime mover (talk) 19:46, 25 January 2009 (UTC)