Converting explicit series terms to summation notation (n ≥ 2)

0:00Let's say that we're told that this sum right over here, where our index starts at 2 and we go all the way to infinity, that this infinite series is negative 8/5 plus 16/7 minus 32/9 plus-- and we just keep going on and on forever. And so what I want to do is to explicitly define what a sub n is here.

0:18So right now we just say, hey, if you take the sum of a sub n from n equals 2 to infinity, it turns out you get this sum right over here. But let's think about what a sub n-- how we can actually define it in terms of n. And I encourage you to pause the video right now and try it on your own.

0:34So the first thing that you might realize is, well, this is the number that we're going to get. Let me write it this way. a sub 2 is equal to negative 8/5. a sub 3 is equal to 16/7. a sub 4 is equal to negative 32/9. And I'm just giving the sign to the number in the numerator.

1:04Negative 8/5 is the same thing as negative 8 over 5. Let me make that a little bit clearer. So I'll make that a little bit clearer. So this is negative 8/5. Obviously, this is positive, so I don't have to really worry about it too much. And then here, I'm just saying negative 32/9, so it's the same thing as negative 32 over 9.

1:25So let's see if we can first find a pattern in the numerator. So when we go from negative 8 to 16, what's happening? Well, we're multiplying by negative 2. Now, to go from 16 to negative 32, we're multiplying by negative 2 again. So you might say, OK, well, whatever we have in the numerator must be a power of negative 2.

1:54And, all right, if you say, well, maybe this is negative 2 squared, well, you know that negative 8 isn't negative 2 squared. Negative 2 squared is equal to positive 4. Negative 8-- this right over here. Negative 8, that is equal to negative 2 to the third power.

2:1016 is equal to negative 2 to the fourth power. Negative 32 is equal to negative 2 to the fifth power. So notice, our exponent on the negative 2 is always going to be one more than our index. Our index is 2, our exponent is 3. Our index is 3, our exponent is 4.

2:36Our index is 4, our exponent is 5. So that gives a sense that at least the numerator is going to be-- whatever our index is, it's going to be-- so let me write this down. So a sub n is equal to-- well, it's going to be negative 2 to whatever index we're at, to that index plus 1 power.

2:59So that's a reasonable way to think about our numerator. Now let's think about our denominators. So over-- So we go from 5, so when n is 2, we're at 5. When n is 3, we're at 7. When n is 4, we're at 9. So notice, 5 is 2 times 2 plus 1. This right over here is 2 times 3 plus 1.

3:32This right over here is 2 times 4 plus 1. And you should just kind of play around with different patterns in your head until you say, hey, well, look, this is increasing by 2 every time. Notice, this increases by 2 every time. But these aren't exactly multiples of 2.

3:51These seem to be off by one more than the multiples of 2, which is a good sign that this is going to be 2 times our index plus 1. So we could write this as 2 times our index plus 1. And we're done. That's what a sub n is. And if we wanted to write this series in sigma notation, we would write this as the sum from n equals 2 to infinity of negative 2 to the n plus 1 power over 2n plus 1.

4:21And that would equal this series right over here.