Wikipedia:Reference desk/Archives/Mathematics/2011 August 19

= August 19 =

Binomial theorem and convergence
Using the fact that $$(1+x)^n = \sum^n_{k=0}\frac{n!}{(n-k)!k!}x^k$$ show that $$\lim_{n \rightarrow \infty}n^\frac{1}{n} = 1$$ How does one get from A to B here? I tried the substitution $$x=-1+n^\frac{1}{n^2}$$ but that didn't make it any easier. Widener (talk) 08:55, 19 August 2011 (UTC)
 * I assume you're meant to do this without logarithms or L'Hôpital's rule. Let
 * $$x=n^{\frac{1}{n}}-1,\, y=\sqrt{\frac{2}{n}}$$
 * Then the binomial theorem implies
 * $$(1+y)^n>1+ny+\frac{n(n-1)}{2}y^2=1+\sqrt{2n}+n-1>n=(1+x)^n$$
 * Take nth roots of both ends to get 1+y>1+x or y>x. The squeeze theorem, using x>0 and y→0 as n→∞, implies x→0 as n→∞.--RDBury (talk) 12:08, 19 August 2011 (UTC)
 * Thanks. Widener (talk) 23:28, 19 August 2011 (UTC)