1/1×2×3×4+1/2×3×4×5+…+1/7×8×9×10=

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/24 02:39:14
1/1×2×3×4+1/2×3×4×5+…+1/7×8×9×10=1/1×2×3×4+1/2×3×4×5+…+1/7×8×9×10=1/1×2×3×4+1/2×3×4×5+…+1/7×8×9×10=发

1/1×2×3×4+1/2×3×4×5+…+1/7×8×9×10=
1/1×2×3×4+1/2×3×4×5+…+1/7×8×9×10=

1/1×2×3×4+1/2×3×4×5+…+1/7×8×9×10=
发现:1/n(n+1)(n+2)(n+3)=[1/n(n+1)(n+2)-1/(n+1)(n+2)(n+3)]/3
所以裂项:
1/1×2×3×4+1/2×3×4×5+...+1/7×8×9×10
=(1/1*2*3-1/2*3*4+1/2*3*4-1/3*4*5+...+1/7*8*9-1/8*9*10)/3
=(1/1*2*3-1/8*9*10)/3
=(1/6-1/720)/3
=119/2160