Limit of Integer to Reciprocal Power/Proof 3

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Theorem

Let $\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation.

Then $\sequence {x_n}$ converges with a limit of $1$.


Proof

Let $n^{1/n} = 1 + a_n$.

The strategy is to:

$(1): \quad$ prove that $a_n > 0$ for $n > 1$
$(2): \quad$ deduce that $n - 1 \ge \dfrac {n \paren {n - 1} } {2!} a_n^2$ for $n > 1$

and hence:

$(3): \quad$ deduce that $0 \le a_n^2 \le \dfrac 2 n$


Let $n > 1$.

Then:

\(\ds n\) \(=\) \(\ds \paren {1 + a_n}^n\)
\(\ds \) \(=\) \(\ds 1 + n a_n + \dfrac {n \paren {n - 1} } {2!} a_n^2\)


This needs considerable tedious hard slog to complete it.
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Sources

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