Thursday, 8 August 2013

analysis - How to show that limnto+inftynfrac1n=1?




I've spent the better part of this day trying to show from first principles that this sequence tends to 1. Could anyone give me an idea of how I can approach this problem?



lim


Answer



You can use \text{AM} \ge \text{GM}.



\frac{1 + 1 + \dots + 1 + \sqrt{n} + \sqrt{n}}{n} \ge n^{1/n} \ge 1




1 - \frac{2}{n} + \frac{2}{\sqrt{n}} \ge n^{1/n} \ge 1


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