lim{(a^n+b^n+c^n)/3}^1/n=?n趋近0答案=（a*b*c)^1/3

Question

lim{(a^n+b^n+c^n)/3}^1/n=?n趋近0 答案=（a*b*c)^1/3

Answer 1

证明：令{(a^n+b^n+c^n)/3}^1/n=u,则lnu=(1/n)*ln((a^n+b^n+c^n)/3)=ln((a^n+b^n+c^n)/3)/n.而limln((a^n+b^n+c^n)/3)/n=lim(3/(a^n+b^n+c^n))*((a^n*lna+b^n*lnb+c^n*lnc)/3)=(3/(1+1+1))*((lna+lnb+lnc)/3)=(lna+lnb+lnc)/3.
即：lim(lnu)=(lna+lnb+lnc)/3.所以limu=e^((lna+lnb+lnc)/3)=(e^(lna+lnb+lnc))^(1/3)==（a*b*c)^1/3.完毕.