Abel's Limit Theorem/Proof 2
Theorem
Let $\ds \sum_{k \mathop = 0}^\infty a_k$ be a convergent series in $\R$.
Then:
- $\ds \lim_{x \mathop \to 1^-} \paren {\sum_{k \mathop = 0}^\infty a_k x^k} = \sum_{k \mathop = 0}^\infty a_k$
where $\ds \lim_{x \mathop \to 1^-}$ denotes the limit from the left.
Proof
 | This article needs to be tidied. In particular: Please attempt to understand the house style. Seriously, I haven't got the patience to tidy up. Please fix formatting and $\LaTeX$ errors and inconsistencies. It may also need to be brought up to our standard house style. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Tidy}} from the code. |
 | This article needs to be linked to other articles. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{MissingLinks}} from the code. |
Since $\ds \sum_{k \mathop = 0}^\infty a_k$ converges and $\cmod{x^k}\le1$ and $\sequence{x^k}$ is decreasing, by Abel's Test for Uniform Convergence $\ds \sum_{k \mathop = 0}^\infty a_kx^k$ converges uniformly on $[0,1]$ and hence to a continuous function by Uniform Limit Theorem.
Also known as
Abel's Limit Theorem is also known just as Abel's Theorem.
However, the latter name has more than one theorem attached to it, so the full name is preferred.
Again, Abel's Limit Theorem can also be found as Abel's Lemma.
However, the latter name is also found attached to a completely different result, so again, it is preferred that it not be used in this context.
Also see
Source of Name
This entry was named for Niels Henrik Abel.