# Proof that the limit as n approaches infinity of n^1/n = 1 ($$\lim_{n \to \infty} n^{1/n} = 1$$)

Here’s an important limit from real analysis that gives quite a few people, including myself, a lot of trouble:
$$\lim_{n \to \infty}n^{1/n} = 1$$

Here is the proof that my Advanced Calculus professor at Western Michigan University, Yuri Ledyaev, gave in class. It uses the binomial expansion.

Proof: Since $$n \in \mathbb{N}$$, for all $$n \geq 2$$ we can write
$$\begin{eqnarray} n^{1/n} &=& 1 + \alpha ~ [where ~ \alpha \geq 0] \nonumber \\ (n^{1/n})^n &=& (1 + \alpha)^n \nonumber \\ n &=& (1 + \alpha)^n \nonumber \\ \end{eqnarray}$$

We want to estimate $$\alpha$$. If $$\alpha$$ is, say, $$0$$, then we’ll have $$n^{1/n} = 1+0$$, meaning the limit we’re after will be $$1$$. The binomial theorem says that
$$\begin{eqnarray} (a+b)^n &=& a^n + na^{n-1}b^1 + \frac{n(n-1)}{2}a^{n-2}b^2 + … +b^n \nonumber \\ &=& \sum\limits_{k=0}^n \binom{n}{k} a^{n-k}b^k\nonumber \\ \end{eqnarray}$$
Therefore,
$$\begin{eqnarray} (1+\alpha)^n &=& 1^n + \binom{n}{1}1^{n-1}\alpha + \binom{n}{2}1^{n-2}\alpha^2 + … +\alpha^n \\[3pt] \nonumber \\ &=& 1^n + n \cdot 1 \cdot \alpha + \frac{n(n-1)}{2} \cdot 1 \cdot \alpha^2 + … + \alpha^n\\[3pt] \nonumber \\ &=& 1 + n\alpha + \frac{n(n-1)}{2}\alpha^2 + … +\alpha^n \\[3pt] \nonumber \\ &>& 1 + n\alpha + \frac{n(n-1)}{2}\alpha^2 \\[3pt] \nonumber \\ &>& 1 + \frac{n(n-1)}{2}\alpha^2 \nonumber \\ \end{eqnarray}$$

So we have
$$\begin{eqnarray} 1+\frac{n(n-1)}{2}\alpha^2 &<& (1+\alpha)^n = n \\[3pt] \nonumber \\ \frac{n(n-1)}{2}\alpha^2 &<& n - 1 < n \\[3pt] \nonumber \\ \alpha^2 &<& \frac{n}{\frac{n(n-1)}{2}} \\[3pt] \nonumber \\ \alpha^2 &<& \frac{2}{n-1} \\[3pt] \nonumber \\ \alpha &<& \sqrt{\frac{2}{n-1}} \nonumber \\ \end{eqnarray}$$ Thus, $$\lim_{n \to \infty}\alpha = 0$$, and $$\lim_{n \to \infty}n^{1/n} = \lim_{n \to \infty}(1+\alpha) = 1+0=1$$. $$\blacksquare$$

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### One Response to Proof that the limit as n approaches infinity of n^1/n = 1 ($$\lim_{n \to \infty} n^{1/n} = 1$$)

1. Alex says:

This is absolutely brilliant, and it helps me to prove that (n!)^(1/n) < (n+1)/2 by induction.
Thanks for posting