Macroeconomics Exam Review 11

Macroeconomics Exam Review 11 - c 2001 Michael Carter All...

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𝛽 = 1, in which case 𝑥 = lim 1 𝑛 = 1 𝑥 = 0 when 0 𝛽 < 1 Therefore 𝛽 𝑛 0 ⇐⇒ 𝛽 < 1 1.103 1. For every 𝑥 ∈ ℜ ( 𝑥 2) 2 0 Expanding 𝑥 2 2 2 𝑥 + 2 0 𝑥 2 + 2 2 2 𝑥 Dividing by 𝑥 𝑥 + 2 𝑥 2 2 for every 𝑥 > 0. Therefore 1 2 ( 𝑥 + 2 𝑥 ) 2 2. Let ( 𝑥 𝑛 ) be the sequence defined in Example 1.64. That is 𝑥 𝑛 = 1 2 ( 𝑥 𝑛 1 + 2 𝑥 𝑛 1 ) Starting from 𝑥 0 = 2, it is clear that 𝑥 𝑛 0 for all 𝑛 . Substituting in 1 2 ( 𝑥 + 2 𝑥 ) 2 𝑥 𝑛 = 1 2 ( 𝑥 𝑛 1 + 2 𝑥 𝑛 1 ) 2 That is 𝑥 𝑛 2 for every 𝑛 . Therefore for every 𝑛 𝑥 𝑛 𝑥 𝑛 +1 = 𝑥
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