Re: [問題] 請問這數列收斂至0還是1？

[plover (+oo)]

※ 引述《age0925 (小立)》之銘言：
: as n→∞, (3/n)^(1/n)=?
: 謝謝解答

It is well-known that (3)^(1/n) -> 1 as n -> oo.  Next, put f(x) = x^(1/x)
and g(x) = log f(x) = 1/x log(x), where x > 0.  By L'Hospital's rule:

 lim  g(x) =  lim  log(x)/x =  lim  (1/x)/1 = 0.  Since h(x) = e^x is
x->oo        x->oo            x->oo

continuous, we have f(x) = e^g(x) = e^0 = 1.  And hence (3/n)^(1/n) -> 1
as n -> oo.

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