Question

The following is an application of harmonic series:

Each book is 1 unit long, then with a stack of n books, we can slide the top book a total of book-lengths off the edge of the table, where Hn is the sequence of partial sums of the Harmonic series. Using this formula, how many books would you need until the top book is a total of 1.5 book-lengths over the edge of the table? [Hint: the answer happens to be less than 13, so just take some partial sums.] Under the hypotheses of the integral test, if an = f(n), then for each positive integer n we have Use this fact to show that ln n le 1 + 1/2 + 1/3 + Â + 1/n le 1 + ln n for all positive integers n. Part (b) shows that 1/2 + ln(n) 1 + 1/2 +1/3 + Â + 1/n is a good approximation for the partial sums of the harmonic series. Use thus approximation to estimate how many books you would need to use to get the top book 5 book-lengths over the edge. [Remember to divide by two in our formula for how far the top book can reach.] Using this same approximation, how many books would you need to get the top book G book-lengths over the edge? How about 8? How about 10? (Your calculator may have trouble with such large numbers!) If you had n = 1087 books available (which is more books th.au the number of atoms estimated in the visible universe), then about how many book-lengths over the edge of the table could you get the top one2? [Hint: I 'm guessing that your calculator can't do computations with numbers that big, so use some properties of logs to first simplify your answer by hand a little bit.]

Show transcribed image text Each book is 1 unit long, then with a stack of n books, we can slide the top book a total of book-lengths off the edge of the table, where Hn is the sequence of partial sums of the Harmonic series. Using this formula, how many books would you need until the top book is a total of 1.5 book-lengths over the edge of the table? [Hint: the answer happens to be less than 13, so just take some partial sums.] Under the hypotheses of the integral test, if an = f(n), then for each positive integer n we have Use this fact to show that ln n le 1 + 1/2 + 1/3 + Â + 1/n le 1 + ln n for all positive integers n. Part (b) shows that 1/2 + ln(n) 1 + 1/2 +1/3 + Â + 1/n is a good approximation for the partial sums of the harmonic series. Use thus approximation to estimate how many books you would need to use to get the top book 5 book-lengths over the edge. [Remember to divide by two in our formula for how far the top book can reach.] Using this same approximation, how many books would you need to get the top book G book-lengths over the edge? How about 8? How about 10? (Your calculator may have trouble with such large numbers!) If you had n = 1087 books available (which is more books th.au the number of atoms estimated in the visible universe), then about how many book-lengths over the edge of the table could you get the top one2? [Hint: I 'm guessing that your calculator can't do computations with numbers that big, so use some properties of logs to first simplify your answer by hand a little bit.]