计算:(1/2+1/3+…+1/2002)(1+1/2+1/3+…+1/2001)−(1+1/2+…+1/2002)(1/2+1/3+…+1/2001)

问题描述:

计算:(

1
2
+
1
3
+…+
1
2002
)(1+
1
2
+
1
3
+…+
1
2001
)−(1+
1
2
+…+
1
2002
)(
1
2
+
1
3
+…+
1
2001
)

假设:a=1+

1
2
+
1
3
+…
1
2001
,b=
1
2
+
1
3
+…+
1
2001

即a-b=1,
∴原式=(b+
1
2002
)a-(a+
1
2002
)b,
=ab+a×
1
2002
-ab-b×
1
2002

=(a-b)×
1
2002

又∵a-b=1
∴原式=
1
2002