Let {bn} be a bounded sequence of nonnegative numbers and r be any number such that 0≤r<1. sn = b1r + b2r^2 + ….. + bnr^n for every index n. Use the monotone convergence theorem to prove that the series {sn} converges.
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Because all these numbers are nonnegative, then it should be clear to you that is an non-decreasing sequence. Now let B be the bound such that . It should also be clear that, . What do you know about a bounded non-decreasing sequence?
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