若数列an=(1+1/n)^n,求证an
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若数列an=(1+1/n)^n,求证an
a_(n+1) = ( 1 + 1/(n+1) ) ^ (n+1)
= ( 1/n + 1/n + ...+ 1/n + 1/(n+1) ) ^ (n+1)
> [ (n+1) ( 1/((n^n*(n+1)) )开(n+1)次方根 ] ^ (n+1) (均值不等式)
= (n+1)^(n+1) * 1/((n^n)*(n+1))
= (n+1)^n / n^n
= ( (n+1)/n ) ^n
= (1+1/n)^n = a_n
再问: 请问如何从第二行到第三行的?具体怎么用的均值不等式,我看不太懂……
= ( 1/n + 1/n + ...+ 1/n + 1/(n+1) ) ^ (n+1)
> [ (n+1) ( 1/((n^n*(n+1)) )开(n+1)次方根 ] ^ (n+1) (均值不等式)
= (n+1)^(n+1) * 1/((n^n)*(n+1))
= (n+1)^n / n^n
= ( (n+1)/n ) ^n
= (1+1/n)^n = a_n
再问: 请问如何从第二行到第三行的?具体怎么用的均值不等式,我看不太懂……
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