Limit of Integer to Reciprocal Power

From ProofWiki
Jump to navigation Jump to search

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 1

From Number to Reciprocal Power is Decreasing we have that the real sequence $\sequence {n^{1/n} }$ is decreasing for $n \ge 3$.

Now, as $n^{1 / n} > 0$ for all positive $n$, it follows that $\sequence {n^{1 / n} }$ is bounded below (by $0$, for a start).

Thus the subsequence of $\sequence {n^{1 / n} }$ consisting of all the terms of $\sequence {n^{1 / n} }$ where $n \ge 3$ is convergent by the Monotone Convergence Theorem (Real Analysis).

Now we need to demonstrate that this limit is in fact $1$.

Let $n^{1 / n} \to l$ as $n \to \infty$.


Having established this, we can investigate the subsequence $\sequence {\paren {2 n}^{1 / {2 n} } }$.

By Limit of Subsequence equals Limit of Real Sequence, this will converge to $l$ also.

From Limit of Root of Positive Real Number, we have that $2^{1 / {2 n} } \to 1$ as $n \to \infty$.

So $n^{1 / {2 n} } \to l$ as $n \to \infty$ by the Combination Theorem for Sequences.

Thus:

$n^{1 / n} = n^{1 / {2 n} } \cdot n^{1 / {2 n} } \to l \cdot l = l^2$

as $n \to \infty$.

So $l^2 = l$, and as $l \ge 1$ the result follows.

$\blacksquare$


Proof 2

We have the definition of the power to a real number:

$\ds n^{1/n} = \map \exp {\frac 1 n \ln n}$

From Powers Drown Logarithms, we have that:

$\ds \lim_{n \mathop \to \infty} \frac 1 n \ln n = 0$

Hence:

$\ds \lim_{n \mathop \to \infty} n^{1/n} = \exp 0 = 1$

and the result.

$\blacksquare$


Proof 3

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:

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