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August 19

Binomial theorem and convergence

Using the fact that $(1+x)^{n}=\sum _{k=0}^{n}{\frac {n!}{(n-k)!k!}}x^{k}$ 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)[reply]

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)[reply]

Thanks. Widener (talk) 23:28, 19 August 2011 (UTC)[reply]