5.5.1

[STARTS AT 4:55]

Instructor: We have the sum from 1 to infinity of negative 1 to the n plus 1 times n divided by 2 to the n. Let's first determine if this is decreasing. All right. So b to the n or b sub n is in over 2 to the n. All right. Now b sub n plus 1 is n plus 1 over 2 to the n plus 1. I would like to start by assuming that it is going to be decreasing and see if we can kind of get some intuition here. 

So this would be we want n plus 1 over 2 to the n plus 1 to be less than or equal to n over 2 to the n. That is true if and only if 2 to the n, n plus 2 to the n, I'm multiplying that across. And because all these terms are going to be positive, they're all greater than or equal to 1, the ns are, and 2 to the whatever power, 2 any power is always positive. I can do this without worrying about the sign there. 

That's 2 to the n plus 1 times n. All right. Let's see what I can do. I can subtract that. Let's rewrite this. 2 times 2 to the n times n is less than n times 2 to the end. Let's keep it in the same order. 2 to the n times n times 2 to the n. If I subtract a 2 to the n times n from both sides, that will be 2 to the n is less than or equal 2. 

That will be one of those. Hold on. I subtract that to be one of those. So be 2 to the n times n. There we go which is true if and only if n is greater than or equal to 1, which is definitely greater than or equal to 1 for all of these circumstances. So ideally, I would now work backwards to show that bn plus 1 is less than or equal to bn. And all the terms are greater than zero, it is decreasing. So this is all part A. 

Part B. Does n over 2 to the n. Does is that limit, in fact going to be zero. Now let's first apply L'Hospital's rule. Actually let's write that clear. Let's put limit over here. And this is because of L'Hospital's rule. 

That will be 1 over the natural log 2 times 2 to the n, which will most certainly. Go to zero. So in this case by the altering series test, sum from 1 to infinity of negative 1 to the n plus 1n over 2 to the n converges.