Question

求极限lim(n趋向于无穷)n*(2^(1/n)-2^(1/(n+1)))

要详细过程

Answer 1

变形=(2^(1/n)-2^(1/(n+1)))/(1/n)

使用洛必达法则
=ln2[2^(1/n)*-1/n^2+2^(1/(n+1))/(n+1)^2]/(-1/n^2)
分子分母同时乘以n^2
=ln2[-2^(1/n)+2^(1/(n+1))*n^2/(n+1)^2]/(-1)
=ln2[-2^(1/n)+2^(1/(n+1))]/(-1)=0

Answer 2

=limn*2^(1/(n+1))*(2^(1/n-1/(n+1))-1)
=limn*1*ln2/n(n+1)
=0
无穷近似值代换a^x-1~xlna