1+1/3-7/12+9/20-11/30+13/42-15/56+17/72

1+1/3=1/2+5/6
前两项不合规律，拆出一个5/6
5/6-7/12+9/20-11/30+13/42-15/56+17/72
分子通项是2n+3
分母通项是(n-1)(n+4)+6 = n*n+3*n+2 用递推关系求出

S=1/2
t=1
for x=1 to 7
t=t*(-1)
s=s+t*(2*x+3)/(x*x+3*x+2)
next
Print "1+1/3-7/12+9/20-11/30+13/42-15/56+17/72=";s

1+1/3-7/12+9/20-11/30+13/42-15/56+17/72
=1+1/3-(3+4)/12+(4+5)/20-(5+6)/30+(6+7)/42-(7+8)/56+(8+9)/72
=1+1/3-1/3-1/4+1/4+1/5-1/5-1/6+1/6+1/7-1/7-1/8+1/8+1/9
=1+1/9
=10/9

因为7/12=(3+4)/3*4=1/3+1/4
9/20=(4+5)/4*5=1/4+1/5
...
13/42=(6+7)/6*7=1/6+1/7
则5/6-7/12+9/20-11/30+13/42-15/56+17/72
=1+1/3-1/3-1/4+1/4+1/5-1/5-1/6+1/6+1/7-1/7-1/8+1/8+1/9
=1+1/9
=10/9