高等数学吧 关注：471,787贴子：3,870,882
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x->+∞
a1^(1/x) = 1+ ln(a1). (1/x) +o(1/x)
a2^(1/x) = 1+ ln(a2). (1/x) +o(1/x)
...
an^(1/x) = 1+ ln(an). (1/x) +o(1/x)
[a1^(1/x)+a2^(1/x)+...+an^(1/x)]/n = 1 + (1/n)ln(a1.a2....an).(1/x) +o(1/x)
lim(x->+∞) { [a1^(1/x)+a2^(1/x)+...+an^(1/x)]/n }^(nx)
=lim(x->+∞) [1 + (1/n)ln(a1.a2....an).(1/x)]^(nx)
=e^[(n/n)ln(a1.a2....an)]
=a1.a2...an
a1^(1/x) = 1+ ln(a1). (1/x) +o(1/x)
a2^(1/x) = 1+ ln(a2). (1/x) +o(1/x)
...
an^(1/x) = 1+ ln(an). (1/x) +o(1/x)
[a1^(1/x)+a2^(1/x)+...+an^(1/x)]/n = 1 + (1/n)ln(a1.a2....an).(1/x) +o(1/x)
lim(x->+∞) { [a1^(1/x)+a2^(1/x)+...+an^(1/x)]/n }^(nx)
=lim(x->+∞) [1 + (1/n)ln(a1.a2....an).(1/x)]^(nx)
=e^[(n/n)ln(a1.a2....an)]
=a1.a2...an
某带魏克星
