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Unformatted text preview: 2. Prove that 1 / â 1 + 1 / â 2 + Â·Â·Â· + 1 / â n â¥ â n for all n â N . We, again, argue using induction. The n = 1 base case follows trivially as both sides are equal to 1 in that case. We then assume that the result holds for n replaced by k-1 and seek to use that to prove the n = k case. Here, by this inductive hypothesis, we have 1 / â 1 + 1 / â 2 + Â·Â·Â· + 1 / â k-1 + 1 / â k â¥ â k-1 + 1 â k . Thus, it will suï¬ce to show that â k-1 + 1 â k â¥ â k. Note that p k ( k-1) â¥ p ( k-1) 2 = k-1 , for k â¥ 2 . Thus, p k ( k-1) + 1 â¥ k in this range. Or â k-1 + 1 â k â¥ â k, which is the desired inequality. 1...
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This note was uploaded on 07/15/2010 for the course MATH 521 taught by Professor Metcalfe during the Spring '10 term at University of North Carolina Wilmington.
- Spring '10