5.1 Approximating Areas [STARTS AT 0:22] PROFESSOR: Write in sigma notation and evaluate the sum of the terms 3 to the i for i equals 1, 2, 3, 4, and 5. Well, the sigma indicates a sum. The sum from i equals 1 through 5 of the terms 3 to the i, that means we have 3 to the 1 plus 3 to the 2 plus 3 to the third 3 to the fourth plus 3 to the fifth. So sigma notation is just a much more compact way of writing this. And that would be 3, 9, 27, 81, and 243. If we sum those, we get 363. Part B-- write the sum in sigma notation 1, 1/4, 1/9, 1/16, and 1/25. We want to sum those. So i equals 1 is generally our starting point. You can start anywhere. But generally, you want to start at 1. Well, if this is the first term, there's our second term. I'm just going to write these here. 1, 2, 3, 4, and 5. So we're going to go to from i equals 1 to 5. And what it looks like is happening is this is 2 squared, this is 3 squared, 4 squared, and 5 squared. So it looks like we have i squared in our denominator, i squared. And it's 1 over i squared. Now, notice i is a dummy variable. The letter i actually never comes up in our system itself, kind of like in part A. We just end up with a number. So saying i squared there just means we're changing the value of i from 1 to 2 to 3 to 4 and all the way to 5. It increments by 1, of course, counting in natural numbers. [ENDS AT 2:30]