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

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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 $\underset{n\to \mathrm{\infty }}{\lim }{n}^{\frac{1}{n}}=1$ 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)