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Question: exercise 19...

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Exercise 1.9

0 A geometric serics will converge precisely when lal < 1. Indeed, in this case, we have lim a by Theorem 1.2, and so using this and (25) implies that calculathon by (1-q) gives (25). n+1 (26) =-. a = lim sn-lim 0 This is one of the few types of series that one can calculate precisely. Here are a few examples: One can calculate geometric series that do not begin at k-0 in two ways: 2 2 (27) or (28) In effect, these ways o f manipulating infinite sums are using some facts about convergent series. Exercise 1.9. Consider carefully the statement in (28). It in essence is an application of the distributive property, but the problem is that the sums are infinite sums! The distributive property is assumed to hold for only finite sums. That is, the distributive property says that Ctl に0 for any n, but then uhy is (28) valid?
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