LimitOfNthRootOfN 2007-12-07 11:53:01 2007-12-09 05:09:47 Example nth root % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \theoremstyle{definition} \newtheorem*{thmplain}{Theorem} The \PMlinkname{$n$th root}{NthRoot} of $n$ tends to 1 as $n$ tends to infinity, i.e. the real number sequence $$\sqrt[1]{1},\, \sqrt[2]{2},\,\sqrt[3]{3},\,\ldots,\,\sqrt[n]{n},\,\ldots$$ converges to the limit \begin{align} \lim_{n\to\infty}\sqrt[n]{n} = 1. \end{align} {\it Proof.}\, If we denote\, $\sqrt[n]{n} := 1+\delta_n$, we may write by the binomial theorem that $$n = (1+\delta_n)^n = 1+{n\choose1}\delta_n+{n\choose2}\delta_n^2+\ldots+{n\choose n}\delta_n^n.$$ This implies, since all \PMlinkescapetext{terms of the right} hand side are positive (when\, $n > 1$), that $$n > {n\choose2}\delta_n^2 = \frac{n(n\!-\!1)}{2!}\delta_n^2,\quad \delta_n^2 < \frac{2}{n-1},\quad 0 < \delta_n < \sqrt{\frac{2}{n-1}},$$ whence\, $\displaystyle\lim_{n\to\infty}\delta_n = 0$.\, Accordingly, $$\lim_{n\to\infty}\sqrt[n]{n} = \lim_{n\to\infty}(1+\delta_n) = 1,$$ Q.E.D.\\ \textbf{Note.}\, (1) follows also from the corollary 3 in the entry growth of exponential function.