(2/3)+(5/9)+(8/27)+(11/81)+(14/243)+...+[(3n-1)/(3^n)]

来源：学生作业帮编辑：作业帮分类：数学作业时间：2024/05/16 10:07:52

(2/3)+(5/9)+(8/27)+(11/81)+(14/243)+...+[(3n-1)/(3^n)]
求该式的值,

设Sn=2/3+5/9+8/27+11/81+14/243+...+(3n-1)/(3^n)
∴1/3Sn=2/9+5/27+8/81+11/243+...+(3n-4)/(3^n)+(3n-1)/3^(n+1)
∴Sn-1/3Sn=2/3Sn
=2/3+3/9+3/27+3/81+.+3/3^n-(3n-1)/3^(n+1)
=2/3-(3n-1)/3^(n+1)+[1/3+1/9+1/27+...+1/3^(n-1)]
=2/3-(3n-1)/3^(n+1)+1/3[1-(1/3)^(n-1)]/(1-1/3)
=2/3-(3n-1)/3^(n+1)+1/2[1-1/3^(n-1)]
=7/6-(3n-1)/3^(n+1)-(9/2)/3^(n+1)
=7/6-(3n+7/2)/3^(n+1)
∴Sn=7/4-(6n+7)/(4×3^n)

证明不等式：(1/n)^n+(2/n)^n+(3/n)^n+.+(n/n)^n lim 9^n+4^n+2/5^n-3^2n-1 n趋于无穷大时计算9^n*(1/27)^n+1*3^n+2 lim(3^2n+5^n)/(1+9^n) 证明(1+2/n)^n>5-2/n（n属于N+,n>=3) 一道数列求和题1/2n+3/4n+5/8n+...+（2n-1）/n*2^n [3n(n+1)+n(n+1)(2n+1)]/6+n(n+2)化简 81^(n+1)÷27^n÷9^(n-1)×3^（n-4） -n^3+8n^2-16n 化简(n+1)(n+2)(n+3) 1\n(n+3)+1\(n+3)(n+6)+1\(n+6)(n+9)=1\2 n+18 n为正整数,求n的值 M=(N-1)×1+(N-2)×2+(N-3)×4+(N-4)×8+(N-5)×16+(N-6)×32+(N-7)×64