求下列数列的极限lim(n趋向于无穷)(1+2^n+3^n+4^n)^1/nlim(n趋向于无穷）1+1/2+1/2^2+…+1/2^n/1+1/4+1/4^2+…+1/4^n

Question

求下列数列的极限
lim(n趋向于无穷)(1+2^n+3^n+4^n)^1/n
lim(n趋向于无穷）1+1/2+1/2^2+…+1/2^n/1+1/4+1/4^2+…+1/4^n

Answer 1

(1)
lim(n-> ∞)(1+2^n+3^n+4^n)^(1/n)
let
L = lim(x->∞)(1+2^x+3^x+4^x)^(1/x)
lnL = lim(x->∞)ln(1+2^x+3^x+4^x) / x ( ∞/∞)
= [(ln2).2^x+(ln3).3^x+(ln4).4^x]/(1+2^x+3^x+4^x)
= [(ln2).(2/4)^x+(ln3).(3/4)^x+(ln4)]/((1/4)^x+(2/4)^x+(3/4)^x+1)
= ln4
L=4
lim(n->∞)(1+2^n+3^n+4^n)^(1/n) = 4
(2)
lim(n->∞)[ 1+(1/2)+(1/2)^2+…+(1/2)^n]/[1+(1/4)+(1/4)^2+…+(1/4)^n]
=lim(n->∞) 3(2^n -1)/(4^n-1)
=lim(n->∞) 3[(2/4)^n -(1/4)^n]/[1-(1/4)^n]
=0