求和:S
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求和:S
![求和:S](/uploads/image/z/3177906-42-6.jpg?t=%E6%B1%82%E5%92%8C%EF%BC%9AS)
因为Sn=
1
2+
3
4+
5
8+
7
16+…+
2n−1
2n,
所以
1
2Sn=
1
4+
3
8+
5
16+…+
2n−3
2n+
2n−1
2n+1,
两式相减得:
1
2Sn=
1
2+
2
4+
2
8+
2
16+…+
2
2n−
2n−1
2n+1=
1
2+
1
2(1−
1
2n−1)
1−
1
2−
2n−1
2n+1,
则Sn=3−
2n+3
2n.
1
2+
3
4+
5
8+
7
16+…+
2n−1
2n,
所以
1
2Sn=
1
4+
3
8+
5
16+…+
2n−3
2n+
2n−1
2n+1,
两式相减得:
1
2Sn=
1
2+
2
4+
2
8+
2
16+…+
2
2n−
2n−1
2n+1=
1
2+
1
2(1−
1
2n−1)
1−
1
2−
2n−1
2n+1,
则Sn=3−
2n+3
2n.