HW#5 - SOLUTIONS TO HOMEWORK # 5 9.2 (a) lim(xn + yn ) =...

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SOLUTIONS TO HOMEWORK # 5 9.2 (a) lim( x n + y n ) = lim x n + lim y n = 3 + 7 = 10. (b) lim 3 y n - x n y 2 n = lim(3 y n - x n ) lim y 2 n = lim3 y n - lim x n (lim y n ) 2 = 3 · 7 - 3 7 2 = 18 49 . 9.3 lim ± a 3 n + 4 a n b 2 n + 1 = lim( a 3 n + 4 a n ) lim( b 2 n + 1) = lim a 3 n + lim 4 a n lim b 2 n + lim1 = (lim a n ) 3 + 4 lim a n (lim b n ) 2 + 1 = a 3 + 4 a b 2 + 1 . 9.5 If lim t n exists, then lim t n +1 = lim ± t 2 n + 2 2 t n = lim( t 2 n + 2) lim(2 t n ) = (lim t n ) 2 + 2 2lim t n , from the limit theorems. Now let lim t n = t . Then lim t n +1 = t since the sequences ( t n ) and ( t n +1 ) differ only by the term t 1 . We have t = t 2 + 2 2 t , 2 t 2 = t 2 + 2 , t 2 = 2 or t = ± 2 . So which of these solutions is the limit? To answer this question we note that all terms of t n are positive . We prove this by Mathematical Induction. Step 1. It is given that t 1 = 1 so the base of induction holds (it is here where we use the information t 1 = 1 provided in the statement of the problem!). Step 2. Assume
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HW#5 - SOLUTIONS TO HOMEWORK # 5 9.2 (a) lim(xn + yn ) =...

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