(1)、1/1×2+1/2×2+1/3×4+…+1/n(n+1)…
=1-1/2+1/2-1/4+1/3-1/4+1/4-1/5+...+1/n-1/(n+1)+.
=3/4+1/3+1/n
=13/12
（2）、Sn=1/2[1-1/3+1/2-1/4+1/3-1/5+1/5-1/7...+1/n-1/(n+2)]
=1/2*4/5
=2/5
(3)、Sn=1/2+2/4+3/8+...+n/2^n
1/2Sn=1/4+2/8+3/16+...+(n-1)/2^n+n/2^(n+1)
两式相减得
1/2Sn=1/2+1/4+1/8+...+1/2^n-n/2^(n+1)
=[1/2(1-1/2^n)]/(1-1/2)-n/2^(n+1)
=1
Sn=2
(4)Sn=1×2∧0+3×2∧1+5×2∧2+…+（2n-1)×2∧(n-1)
2Sn=1×2∧1+3×2∧2+5×2∧3+…+（2n-3)×2∧(n-1)+（2n-1)×2∧n
两式相减得
-Sn=1×2∧0+2×2∧1+2×2∧2+2×2∧3+…+2×2∧(n-1)-（2n-1)×2∧n
=1+2^2+2^3+...2^n-（2n-1)×2∧n
=-3+2^n-n2^(n+1)
Sn=3-2^n+n2^(n+1)
(5)an=(2n+1)×2∧n,
数列2∧n/(2n-1)an=2^2n×(2n+1)/(2n-1)
=2^2n×(1+2/(2n-1))
=2^2n+2^(2n+1)/(2n-1)
实在不想写了,就那个思路,自己想想就好了