Question

设a>b>c>0,求lim(an+bn+cn)1/n

设a>b>c>0,求lim(an+bn+cn)1/n

Answer 1

c < lim(n→∞) (a^n+b^n+c^n)^1/n < a
lim(n→0) (a^n+b^n+c^n)^1/n = +∞
－－－－－－－－－
解析：
A = (c^n+c^n+c^n)^(1/n) = (3c^n)^(1/n) = c*3^(1/n)
B = (a^n+b^n+c^n)^1/n
C = (a^n+a^n+a^n)^(1/n) = (3a^n)^(1/n) = a*3^(1/n)
所以 A<B<C

当 n→∞ 时，
lim(n→∞) A = lim [c*3^(1/n)] = c
lim(n→∞) C = lim [a*3^(1/n)] = a
因此
c < lim(n→∞) B < a

当 n→0 时，
lim(n→0) A = lim [c*3^(1/n)] = +∞
因为 A<B
所以
lim(n→0) B = +∞