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Unformatted text preview: ( x nx n +1 ) 2 = x 2 n +110 = ⇒ x 2 n +110 ≥ = ⇒ x 2 n +1 2 ≥ 5 = ⇒ x n +1 2 ≥ 5 x n +1 since ∀ n ∈ N ,x n > = ⇒ x n +1 ≥ x n +1 2 + 5 x n +1 = ⇒ x n +1 ≥ x n +2 Therefore x n +1 ≥ x n +2 for all n ∈ N . Therefore x ( n ) is ultimately decreasing. Therefore < x n < x 2 for all n ∈ N so that ( x n ) is bounded as well as monotone. Therefore by the Monotone Convergence Theorem, ( x n ) is convergent. 1 To fnd lim( x n ), let x = lim( x n ) = lim( x n +1 ) by virtue oF the Fact that any tail oF ( x n ) also converges to x . Then, x = lim( x n +1 ) = lim( x n ) 2 + 5 lim( x n ) = ⇒ x = x 2 + 5 x = ⇒ 2 x 2 = x 2 + 10 = ⇒ x 2 = 10 = ⇒ x = √ 10 since ∀ n ∈ N ,x n > ThereFore lim( x n ) = √ 10. Q.E.D. 2...
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 Spring '08
 Akhmedov,A
 #, Xn, Dominated convergence theorem, Monotone convergence theorem, David Joseph Stith

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