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Unformatted text preview: uous. Exercise 66 : Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a n = n + 1 n Solution : The sequence is increasing. To see this, we need to show that a n +1 > a n for all n . Plugging into the formula gives that n + 1 + 1 n + 1 > n + 1 n . This is true if and only if 1 + 1 n + 1 > 1 n , or 1 > 1 n-1 n + 1 . But 1 n-1 n +1 = 1 n ( n +1) , so we can rewrite the last line as 1 > 1 n ( n + 1) . This is clearly true if n ≥ 1, so the original inequality a n +1 > a n must also be true. Thus, the sequence a n is increasing. The sequence is not bounded; let M be any real number, and let n be an integer greater than or equal to M . Then a n = n + 1 n > n ≥ M, so a n > M . As M was arbitrary, there cannot be any number that bounds the sequence above....
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- Spring '08
- Math, lim, Order theory