Sums, Approximations, and Asymptotics II 1.2 Bounding a Sum with an Integral We need to know more about harmonic sums to determine what can be done with a large number of blocks. Unfortunately, there is no closed form for H, But, on the bright side we can get good lower and upper bounds on harmonic sums using a general technique involving integration that applies to many other sums as well Here's how it works. First, we imagine a bar graph where the area of the k-th bar is equal to the k-th term in the sum. In particular, each bar has width 1 and height equal to the value of the k-th term. For example, the bar graph corresponding to the harmonic sum 111 2+3 is shown below 1 1/2 1 01234 Now the value of the sum is equal to the area of the bars, and we can estimate the area of the bars using integration. Suppose we draw a smooth curve that runs just below the bars; in fact, there's a natural choice: the curve described by the function y= 1/(a+1) 1/2 =1/(x+1) 14 Sums, Approximations, and Asymptotics II 1.2 Bounding a Sum with an Integral We need to know more about harmonic sums to determine what can be done with a large number of blocks. Unfortunately, there is no closed form for Hn. But, on the bright side, we can get good lower and upper bounds on harmonic sums using a general technique involving integration that applies to many other sums as well. Here’s how it works. First, we imagine a bar graph where the area of the k­th bar is equal to the k­th term in the sum. In particular, each bar has width 1 and height equal to the value of the k­th term. For example, the bar graph corresponding to the harmonic sum 1 1 1 1 Hn = + + + . . . + 1 2 3 n is shown below. 6 1 1/2 1 1 1 1 . . . 1 2 3 4 - 0 1 2 3 4 n − 1 n Now the value of the sum is equal to the area of the bars, and we can estimate the area of the bars using integration. Suppose we draw a smooth curve that runs just below the bars; in fact, there’s a natural choice: the curve described by the function y = 1/(x + 1). - 6 1 1 1 2 1 3 1 4 1 1/2 y = 1/(x + 1) 0 1 2 3 4 n − 1 n