做个题lim[1/2×5 +1/5×8+1/8×11.+1/(3n-1)(3n+2) n→∞
问题描述：
做个题lim[1/2×5 +1/5×8+1/8×11.+1/(3n-1)(3n+2) n→∞
答
lim{1/(2×5) +1/(5×8)+1/(8×11)....+1/[(3n-1)(3n+2)]} n→∞??
1/(2×5) +1/(5×8)+1/(8×11)....+1/[(3n-1)(3n+2)]
=(1/3)[(1/2-1/5)+(1/5-1/8)+...+[1/(3n-1)-1/(3n+2)]
=(1/3)[1/2-1/(3n+2)]
lim[n→∞]{1/(2×5) +1/(5×8)+1/(8×11)....+1/[(3n-1)(3n+2)]}
=lim[n→∞](1/3)[1/2-1/(3n+2)]
=1/6.
答
1/2×5 +1/5×8+1/8×11.+1/(3n-1)(3n+2)=1/3*（1/2-1/5+1/5-1/8+...+1/(3n-1)-1/(3n+2)）
=1/3*(1/2-1/(3n+2))
所以lim(n→∞)[1/2×5 +1/5×8+1/8×11.+1/(3n-1)(3n+2) ]
=lim(n→∞)[1/3*(1/2-1/(3n+2))]
=1/3*(lim(n→∞)1/2-lim(n→∞)(1/(3n=2))
=1/3*(1/2-0)
=1/6