Question

求极限 lim (n→∞)(2^n+3^n+5^n)^(1/n) = ？ 要详细过程。3Q

Answer 1

lim [n√(2^n+3^n+5^n)]e^{lim [(1/n)*ln(2^n+3^n+5^n)]}
对lim [(1/n)*ln(2^n+3^n+5^n)]用L'HOPITAL法则

lim [(1/n)*ln(2^n+3^n+5^n)]
=lim [(ln2*2^n+ln3*3^n+ln5*5^n)/(2^n+3^n+5^n)]
=lim {[ln2*(2/5)^n+ln3*(3/5)^n+ln5]/[(2/5)^n+(3/5)^n+1]}
当n→∞时(2/5)^n和(3/5)^n均趋向于0
故lim [1/n *ln(2^n+3^n+5^n)] = ln5

lim (n√2^n+3^n+5^n)
=e^{lim [(1/n)*ln(2^n+3^n+5^n)]}
=e^ln5
=5

Answer 2

lim<n→∞>(2^n+3^n+5^n)^(1/n)
=lim<n→∞>{(5^n)*[1+(2^n+3^n/5^n)]}^(1/n)
=5*lim<n→∞>[1+(2^n+3^n/5^n)]^(1/n)
=5*lim<n→∞>{[1+(2^n+3^n/5^n)]^(5^n/2^n+3^n)}^(2^n+3^n/5^n)^(1/n)
=5*lim<n→∞>e^[(2^n+3^n)/(n*5^n)]
=5*e^[lim<n→∞>(2^n*ln2+3^n*ln3)/(5^n+n*5^n*ln5)]
=5*e^0
=5*1
=5

其实这题就是提取个5^n出来就明显了

Answer 3

应该是log(5)6+1吧。2^n+3^n+5^n=(5^n)^[log(5)6+1]，所以lim (n→∞)(2^n+3^n+5^n)^(1/n) =log(5)6+1。