greatest series

which term of the following series is largest 1, 2^(1/2) , 3^(1/3) ....................... , n^(1/n).

Note by Jinay Patel
4 years, 9 months ago

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We have $\lim_{n\to\infty} \big(1 + \tfrac{1}{n}\big)^n \; = \; e$ and we can also show that the sequence $$\big(1+\tfrac{1}{n}\big)^n$$ is an increasing one. Thus we deduce that $$\big(1+\tfrac{1}{n}\big)^n \le e \le n$$ for all $$n \ge 3$$. Thus $\begin{array}{rcl} \ln n & \ge & n\ln\big(1+ \tfrac{1}{n}\big) \\ (n+1) \ln n & \ge & n\ln(n+1) \\ \tfrac{1}{n}\ln n & \ge & \tfrac{1}{n+1}\ln(n+1) \end{array}$ for all $$n \ge 3$$. This tells us that $$n^{\frac{1}{n}} \, \ge \, (n+1)^{\frac{1}{n+1}}$$ for $$n \ge 3$$.

Since $$1 < 2^{\frac12} < 3^{\frac13}$$, we deduce that $$3^{\frac13}$$ is the largest value in the sequence.

- 4 years, 9 months ago

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