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Macroeconomics Exam Review 58

Macroeconomics Exam Review 58 - Solutions for Foundations...

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since the term in brackets is strictly greater than 1 for any 𝑥 > 0. Similarly 𝑒 𝑥 𝑥 = ( 𝑒 𝑥/ ( 𝑛 +1) ) 𝑛 𝑒 𝑥/ ( 𝑛 +1) ( 𝑛 + 1) 𝑛 ( 𝑥 𝑛 +1 ) 𝑛 = 1 ( 𝑛 + 1) 𝑛 ( 𝑒 𝑥/ ( 𝑛 +1) 𝑥/ ( 𝑛 + 1) ) 𝑛 𝑒 𝑥/ ( 𝑛 +1) → ∞ 2.8 Assume that 𝑆 ⊆ ℜ is compact. Then 𝑆 is bounded (Proposition 1.1), and there exists 𝑀 such that 𝑥 ∣ ≤ 𝑀 for every 𝑥 𝑆 . For all 𝑛 𝑚 2 𝑀 𝑓 𝑛 ( 𝑥 ) 𝑓 𝑚 ( 𝑥 ) = 𝑛 𝑘 = 𝑚 +1 𝑥 𝑘 𝑘 ! 𝑥 𝑚 +1 ( 𝑚 + 1)! 𝑛 𝑚 𝑘 =0 ( 𝑥 𝑚 ) 𝑘 𝑀 𝑚 +1 ( 𝑚 + 1)! 𝑛 𝑚 𝑘 =0 ( 𝑀 𝑚 ) 𝑘 𝑀 𝑚 +1 ( 𝑚 + 1)! ( 1 + 1 2 + 1 4 + ⋅ ⋅ ⋅ + ( 1 2 ) 𝑛 𝑚 ) 2 𝑀 𝑚 +1 ( 𝑚 + 1)! 2 ( 𝑀 𝑚 + 1 ) 𝑚 +1 ( 1 2 ) 𝑚 by Exercise 1.206. Therefore 𝑓 𝑛 converges to 𝑓 for all 𝑥 𝑆 . 2.9 This is a special case of Example 2.8. For any 𝑓, 𝑔 𝐹 ( 𝑋 ), define ( 𝑓 + 𝑔 ) = 𝑓 ( 𝑥 ) + 𝑔 ( 𝑥 ) ( 𝛼𝑓 )( 𝑥 ) = 𝛼𝑓 ( 𝑥 ) With these definitions 𝑓 + 𝑔 and
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