Harmonic Series Divergence Problem: If you stack a deck of cards so that they just

Chapter 14, Problem 4

(choose chapter or problem)

Harmonic Series Divergence Problem: If you stack a deck of cards so that they just barely balance, the top card overhangs by the deck length, the second card overhangs by the deck length, the third card overhangs by the deck length, and so on (Figure 14-3b). Figure 14-3b The total overhang for n cards is thus a partial sum of the harmonic series a. The figure indicates that the total overhang for three cards is greater than the length of the deck. Show numerically that this is true. b. How many cards would you have to stack in order for the total overhang to exceed two deck lengths? c. What would the total overhang be for a standard 52-card deck? Surprising? d. Associate the terms of the harmonic series this way: The terms are grouped into groups of 1, 2, 4, 8, 16, . . . terms. Show that each group of terms is greater than or equal to . How does this fact allow you to conclude that the partial sums of a harmonic series diverge and can get larger than any real number?