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Unformatted text preview: MAT 319/320 Solutions of Practice Midterm II Problem 1. Define a sequence ( x n ) by : x = 1 and x n +1 = 1 3 x n + 1 . Does ( x n ) have a limit? If yes, what is this limit? Proof. First, notice that if the sequence has a limit l , then it must satisfy the equation : l = 1 3 .l + 1 , thus the only possible limit is l = 3 2 . Let’s show that this sequence is monotone and bounded, and therefore it will be convergent. 1. Claim: the sequence is bounded above by 3 2 . Indeed, by induction we get that: x lessorequalslant 3/2 , and if x n lessorequalslant 3/2 then we deduce that x n +1 lessorequalslant 1 3 . 3 2 + 1 = 3/2 , so we are done. 2. Claim: the sequence is increasing: this comes from the fact that ( x lessorequalslant 3/2) ⇒ 1 3 x + 1 greaterorequalslant x . Therefore the sequence is converging to 3/2 . square Problem 2. Is the infinite series ∑ n =1 + ∞ 1 n + n √ convergent? (If yes, you do not need to find the value of the limit)....
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This note was uploaded on 01/31/2011 for the course AMS 319 taught by Professor Mark during the Fall '10 term at SUNY Stony Brook.
- Fall '10