70 In the situation of exercise 69 if you have two different cards out of ten

70 in the situation of exercise 69 if you have two

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70. In the situation of exercise 69, if you have two different cards out of ten, the average number of cards to get a third distinct
648 CHAPTER 8 . . Infinite Series 8-38 card is k = 1 8 k 2 k 1 10 k ; show that this is a convergent series with sum 10 8 . 71. Extend the results of exercises 69 and 70 to find the average number of cards you need to obtain to complete the set of ten different cards. 72. Compute the ratio of cards obtained to cards in the set in exercise 71. That is, for a set of 10 cards, on the average you need to obtain times 10 cards to complete the set. 73. Generalize exercises 71 and 72 in the case of n cards in the set ( n > 2). 74. Use the divergence of the harmonic series to state the unfortu- nate fact about the ratio of cards obtained to cards in the set as n increases. EXPLORATORY EXERCISES 1. Numerically investigate the p -series k = 1 1 k 0 . 9 and k = 1 1 k 1 . 1 and for other values of p close to 1. Can you distinguish convergent from divergent series numerically? 2. You know that k = 2 1 k diverges. This is the “smallest” p -series that diverges, in the sense that 1 k < 1 k p for p < 1. Show that k = 2 1 k ln k diverges and 1 k ln k < 1 k . Show that k = 2 1 k ln k ln(ln k ) diverges and 1 k ln k ln(ln k ) < 1 k ln k . Find a series such that k = 2 a k diverges and a k < 1 k ln k ln(ln k ) . Is there a smallest divergent series? 3. In this exercise, you explore the convergence of the infinite product P = 2 1 / 4 3 1 / 9 4 1 / 6 ··· . This can be written in the form P = k = 2 k 1 / k 2 . For the partial product P n = n k = 2 k 1 / k 2 , use the natural logarithm to write P n = e ln P n = e ln[2 1 / 4 3 1 / 9 4 1 / 16 ··· n 1 / n 2 ] = e Sn , where S n = ln [2 1 / 4 3 1 / 9 4 1 / 16 ··· n 1 / n 2 ] = 1 4 ln 2 + 1 9 ln 3 + 1 16 ln 4 + ··· + 1 n 2 ln n . By comparing to an appropriate integral and showing that the intergal converges, show that { S n } converges. Show that { P n } converges to a number between 2.33 and 2.39. Use a CAS or calculator to compute P n for large n and see how accurate the computation is. 4. Define a function f ( x ) in the following way for 0 x 1. Write out the binary expansion of x . That is, x = a 1 2 + a 2 4 + a 3 8 + ··· where each a i is either 0 or 1. Prove that this infinite series converges. Then f ( x ) is the corresponding ternary expansion, given by f ( x ) = a 1 3 + a 2 9 + a 3 27 + ··· Prove that this series converges. There is a subtle issue here of whether the function is well defined or not. Show that 1 2 can be written with a 1 = 1 and a k = 0 for k 2 and also with a 1 = 0 and a k = 1 for k 2. Show that you get dif- ferent values of f ( x ) with different representations. In such cases, we choose the representation with as few 1’s as possi- ble. Show that f (2 x ) = 3 f ( x ) and f ( x + 1 2 ) = 1 3 + f ( x ) for 0 x 1 2 . Use these facts to compute 1 0 f ( x ) dx . Generalize the result for any base n conversion f ( x ) = a 1 n + a 2 n 2 + a 3 n 3 + ··· , where n is an integer greater than 1.

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