Abel’s Theorem

Posted on May 25, 2016 by

Abel’s Theorem is a useful theorem in analysis.

Let (a_k) be any sequence in \mathbb{R} or \mathbb{C}. Let G_a(z)=\sum_{k=0}^\infty a_k z^k. Suppose that the series \sum_{k=0}^\infty a_k converges. Then
\lim_{z\to 1^-}G_a(z)=\sum_{k=0}^\infty a_k, where z is real, or more generally, lies within any Stolz angle, i.e.\ a region of the open unit disk where |1-z|\leq M(1-|z|) for some M.

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

Gravatar
WordPress.com Logo

You are commenting using your WordPress.com account. ( Log Out /  Change )

Google photo

You are commenting using your Google account. ( Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. ( Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. ( Log Out /  Change )

Cancel

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.