Infinite products An infinite product P = a1 a2 a3c, which
Chapter 8, Problem 88(choose chapter or problem)
Infinite products An infinite product P = a1 a2 a3c, which is denoted q _ k = 1 ak, is the limit of the sequence of partial products 5a1, a1 a2, a1 a2 a3, c6. Assume that ak 7 0 for all k. a. Show that the infinite product converges (which means its sequence of partial products converges) provided the series a _ k = 1 ln ak converges. b. Consider the infinite product P = q _ k = 2 a1 - 1 k2 b = 3 4 # 8 9 # 15 16 # 24 25 g. Write out the first few terms of the sequence of partial products, Pn = q n k = 2 a1 - 1 k2 b 1for example, P2 = 34 , P3 = 23 2. Write out enough terms to determine the value of P = lim nS_ Pn. c. Use the results of parts (a) and (b) to evaluate the series a _ k = 2 ln a1 - 1 k2 b.
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