1/3+1/8+1/15+……+1/120=?答对有奖
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1/3+1/8+1/15+……+1/120=?答对有奖
1/3+1/8+1/15+……+1/120=?答对有奖
1/3+1/8+1/15+……+1/120=?答对有奖
1/3+1/8+1/15+.+1/120
=1/(1*3)+1/(2*4)+1/(3*5)+……+1/(10*12)
=1/2*[(1/1-1/3)+(1/2-1/4)+(1/3-1/5)+……+(1/10-1/12)]
=1/2*(1+1/2-1/11-1/12)
=1/2*175/132
=175/264
3=2*2-1=(2-1)*(2+1)
8=3*3-1=(3-1)*(3+1)
同理120=11*11-1=(11-1)*(11+1)
1/3=1/(2-1)*(2+1)=(1/2)*(1/(2-1)-1/(2+1))=
(1/2)*(1-1/3)
1/8=1/(3-1)*(3+1)=(1/2)*(1/(3-1)-1/(3+1))=
(1/2)*(1...
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3=2*2-1=(2-1)*(2+1)
8=3*3-1=(3-1)*(3+1)
同理120=11*11-1=(11-1)*(11+1)
1/3=1/(2-1)*(2+1)=(1/2)*(1/(2-1)-1/(2+1))=
(1/2)*(1-1/3)
1/8=1/(3-1)*(3+1)=(1/2)*(1/(3-1)-1/(3+1))=
(1/2)*(1/2-1/4)
1/15=1/(4-1)*(4+1)=(1/2)*(1/(4-1)-1/(4+1))=
(1/2)*(1/3-1/5)
1/120=1/(11-1)*(11+1)=(1/2)*(1/(11-1)-1/(11+1))=
(1/2)*(1/10-1/12)
代入得
1/3+1/8+1/15+……+1/120=(1/2)*(1-1/3+1/2-1/4+1/3-1/5+
……+1/9-1/11+1/10-1/12)
=(1/2)*(1+1/2-1/11-1/12)=175/264
关键是 1/(a-b)*(a+b) =(1/2b)*[1/(a-b)-1/(a+b)]的应用。可以解决这一类题。
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