题目

求函数的极限：lim(1^n+2^n+3^n+4^n)^1/n,当n→∞时的极限.(不用夹逼准则解)

提问时间：2021-03-25

答案

n→∞
lim(1^n+2^n+3^n+4^n)^(1/n)
=e^lim[(1/n)*ln(1^n+2^n+3^n+4^n)]
下面求lim[(1/n)*ln(1^n+2^n+3^n+4^n)]
=lim(1/n)*ln{(4^n)*[(1/4)^n+(2/4)^n+(3/4)^n+1]}
=lim(1/n)*{nln4+ln[1+(1/4)^n+(2/4)^n+(3/4)^n]}
这里ln[1+(1/4)^n+(2/4)^n+(3/4)^n]等价于(1/4)^n+(2/4)^n+(3/4)^n
=ln4+lim[(1/4)^n+(2/4)^n+(3/4)^n]/n
=ln4
所以最后结果为e^ln4=4