Number to Reciprocal Power is Decreasing

Theorem

The real sequence $\sequence {n^{1/n} }$ is decreasing for $n \ge 3$.

We want to show that $\paren {n + 1}^{1 / \paren {n + 1} } \le n^{1/n}$.

Thus:

$\ds \paren {n + 1}^{1 / \paren {n + 1} }$

$\le$

$\ds n^{1/n}$

$\ds \leadstoandfrom \ \ $

$\ds \paren {n + 1}^n$

$\le$

$\ds n^{n + 1}$

raising both sides to the power of $n \paren {n + 1}$

$\ds \leadstoandfrom \ \ $

$\ds \paren {n \paren {1 + \frac 1 n} }^n$

$\le$

$\ds n^{n + 1}$

$\ds \leadstoandfrom \ \ $

$\ds \paren {1 + \frac 1 n}^n$

$\le$

$\ds \frac {n^{n + 1} } {n^n} = n$

$\paren {1 + \dfrac 1 n}^n < 3$

Thus the reversible chain of implication can be invoked and we see that $\paren {n + 1}^{1 / \paren {n + 1} } \le n^{1/n}$ when $n \ge 3$.

So $\sequence {n^{1 / n} }$ is decreasing for $n \ge 3$.

$\blacksquare$