3c S 1 2 1 Assume k S for S k1 S k Then S 1 2 S 2 275 S 3 3125 S 4 33125 The

3c s 1 2 1 assume k s for s k1 s k then s 1 2 s 2 275

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3c) S 1 =2 1 Assume k S for S k+1 S k Then S 1 =2, S 2 =2.75, S 3 =3.125, S 4 =3.3125 The sequence is monotone because it is always increasing for each N ∈ℕ . The monotone sequence is bounded above, and converges to 3.5. The limit of the sequence is 3.5. 4a) a n = (−1) 𝑛 𝑛 , { -1, ½, - 1/3, ¼, …} 4b) a n = n or a n = -n 4c) a n = (−1) 𝑛 , n ∈ℕ is bounded from -1 to 1 but does not converge. It has two limits as n approaches + 6a) Suppose a n+1 = a n +2 , ∀ n ∈ℕ . a 1 =2, a 2 =3, a 3 =5, a 4 =7. The sequence is monotone because it is increasing and a n is a multiple of n, for some k ∈ ℝ ,, then a n is increasing with k c n is monotone for k ∈ ℝ 6b) Suppose a n+1 = a n +2 be an increasing sequence because a 1 =1 and a n+1 a n , for any n ∈ℕ .
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Let b n = 𝑛+1 𝑛 2 for any n ∈ℕ . b 1 =2, b 2 =4, b 3 = 4 9 , b 4 = 5 16 . The sequence is a decreasing sequence. If C n= ? 𝑛 ? 𝑛 , then C 1 =1/2, C 2 = 4, C 3 =11.25, and C 4 =22.4. C n is strictly increasing monotone. Similarly if a n = 𝑛+1 𝑛 2 and b n+1 = b n +2, then C n is strictly decreasing and is monotone. 9) We can begin by squaring to get rid of the square root, then we have: (S n+1 ) 2 -x= ( 𝑆 𝑛 2 +𝑥 2𝑆 𝑛 ) 2 -x 𝑆 𝑛 4 +2𝑆 𝑛 𝑥+𝑥 2 −4𝑆 𝑛 2 𝑥 4𝑆 𝑛 2 𝑆 𝑛 4 −2𝑆 𝑛 𝑥+𝑥 2 4𝑆 𝑛 2 ( 𝑆 𝑛 2 −𝑥 2𝑆 𝑛 ) 2 0 So S n ≥ √𝑥 for any n 2, so S n is bounded below by √𝑥 .
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  • Tn, Sn, lim sn

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