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781hw11

# 781hw11 - 18.781 Fall 2007 Problem Set 11 Due WEDNESDAY...

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18.781, Fall 2007 Problem Set 11 Due: WEDNESDAY, November 28 These problems all relate to the proof that there are infinitely many primes in an arithmetic progression. While this is covered in Niven, Zucker- man, and Montgomery, our proof differs from theirs and notes are available in the ”Solutions and Handouts” section of the course webpage. 1. We will say that an infinite product n =1 a n converges if the limit of the partial products P N = N n =1 a n converges as N → ∞ and is not 0. (That’s the first approximation to the definition. In practice, we want to allow a finite number of the factors a n in the product to be 0. Suppose N 0 is the largest index with a N 0 = 0. Then we will say the product converges if P N = N n = N 0 +1 a n converges as N → ∞ .) (a) Give an example of an infinite product whose partial products converge to 0. (Hence, we don’t regard it as an infinite product.) (b) Show that any convergent infinite product can be written in the form n =1 (1 + b n ) with lim n →∞ b n = 0

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