3.1.7 Integral Method Video
PROFESSOR: So we figured out that you can get the book stack, the overhang of the books, to be half the harmonic sum. With n books, you can get out Hn over 2 where Hn is this harmonic sum, or harmonic number. The question is, how are we going to estimate or calculate what this sum is? Now it turns out there is no simple formula for exactly what this sum is, but there is a simple formula that estimates it quite accurately. And we get the estimation by bounding the sum by integrals. And so let's look at this integral method for estimating sums. Remember what I wanted was the sum of 1 plus 1/2 plus 1/3 down to 1 over n. So let's form some unit width rectangles of heights equal to the amount that I want. So here's a rectangle of height 1, rectangle of height 1/2, rectangle of height 1/3. I'm going out here, if you actually count to 8, but let's propose that this is a height 1 over n. And what I know is that the total area of these rectangles is actually eq...