Sn=1/(2^2-1)+1/(4^2-1)+1/(6^2-1)+.+1/[(2n)^2-1]求和

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Sn=1/(2^2-1)+1/(4^2-1)+1/(6^2-1)+.+1/[(2n)^2-1]求和Sn=1/(2^2-1)+1/(4^2-1)+1/(6^2-1)+.+1/[(2n)^2-1]求和Sn

Sn=1/(2^2-1)+1/(4^2-1)+1/(6^2-1)+.+1/[(2n)^2-1]求和
Sn=1/(2^2-1)+1/(4^2-1)+1/(6^2-1)+.+1/[(2n)^2-1]
求和

Sn=1/(2^2-1)+1/(4^2-1)+1/(6^2-1)+.+1/[(2n)^2-1]求和
Sn=1/(2^2-1)+1/(4^2-1)+1/(6^2-1)+.+1/[(2n)^2-1]
=1/(2+1)(2-1)+1/(4+1)(4-1)+1/(6+1)(6-1)+.+1/(2n+1)(2n-1)
=1/(3*1)+1/(5*3)+1/(7*5)+.+1/(2n+1)(2n-1)
因为1/(2n+1)(2n-1)=0.5*[1/(2n-1)-1/(2n+1)]
所以
Sn=0.5*[1-1/3+1/3-1/5+1/5-1/7+.+1/(2n-1)-1/(2n+1)]
=0.5*[1-1/(2n +1 )]
=n/(2n+1)

Sn的通项:1/[(2n)^2-1]=1/(4n^2-1)=1/(2n+1)(2n-1)
令2n-1=t,2n+1=t+2
1/(2n+1)(2n-1)=1/t(t+2)=(1/2)[1/t-1/(t+2)]
累加：SN=n/(2n+1)

Sn=1/(2^2-1)+1/(4^2-1)+1/(6^2-1)+.....+1/[(2n)^2-1]
=1/2[1-1/3+1/3-1/5+......+1/(2n-1)-1/(2n+1)]
=1/2[1-1/(2n+1)]
=n/(2n+1)

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