Limit of Integer to Reciprocal Power/Proof 2

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

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.

$\blacksquare$

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.7 \ (4)$