5.1.3

[STARTS AT 11:53]

Instructor: Part C. an is defined by a equals 1, and an equals an minus 1 divided by 2. All right, so if we start playing around, as I like to say, first term is 1. The second term is 1/2. Third term is 1/4 because we're dividing by 2 each time. 

We need to determine what is going on with this. Let's see, we still have our same results. All right, so first, an plus 1 is 1/2 of an. It's cutting in half each time. Which means, since that is less than or equal to a sub n, that is definitely decreasing. 

So since we are monotonic, we need to see if it's bounded. Well, again, if you notice this, this is actually equal to a sub n is 1 over 2 to the n-- 2 to the n minus 1, actually, to adjust for that first term being 1. So if you evaluate that at 1, that would be 1 over 2 to the 0, which is 1. Evaluating that at 2, that would be 1/2, et cetera. 

So we know that a sub n is greater than or equal to 0. It approaches 0, but it doesn't quite get there. Well, since a sub n is equal to 1 over 2 to the n minus 1, we know that that approaches 0 just based on our limit properties.

[ENDS AT 13:42]