Sergei Yakovenko's blog: on Math and Teaching

Wednesday, December 28, 2011

Lectures 8-9, December 13 and 20

Limits of sequences

We spend some time considering different flavors of “limit behavior”: stabilization, approximate stabilization etc. A sequence is called converging, if it \varepsilon-stabilizes for any positive accuracy \varepsilon>0.

To show that passing to a limit “respects” arithmetic operations, we need to work out a bit of “interval arithmetic” with a special attention to the division which may cause problems.

We discuss the weaker notion of a partial limit (accumulation point) and study under what assumptions a unique partial limit is the genuine limit.

Finally, we show that monotone bounded sequences always converge. This is one of the most powerful tools to show that the limit exists when it is not possible to compute it explicitly.

The lecture notes (in pdf) are available here.

Leave a Comment »

No comments yet.

RSS feed for comments on this post. TrackBack URI

Leave a Reply

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

WordPress.com Logo

You are commenting using your WordPress.com 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 )

Google+ photo

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

Connecting to %s

Create a free website or blog at WordPress.com.

%d bloggers like this: