求和：Sn=1/2^2-1+1/4^2-1+……+1/(2n)^2-1

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求和：Sn=1/2^2-1+1/4^2-1+……+1/(2n)^2-1求和：Sn=1/2^2-1+1/4^2-1+……+1/(2n)^2-1求和：Sn=1/2^2-1+1/4^2-1+……+1/(2n

求和：Sn=1/2^2-1+1/4^2-1+……+1/(2n)^2-1
求和：Sn=1/2^2-1+1/4^2-1+……+1/(2n)^2-1

求和：Sn=1/2^2-1+1/4^2-1+……+1/(2n)^2-1
Sn=1/2^2-1+1/4^2-1+……+1/(2n)^2-1
=1/(1*3)+1/(3*5)+1/(5*7)+……+1/(2n-3)(2n-1)-1/(2n-1)(2n+1)
=(1/2)[1-1/3+1/3-1/5+……+1/(2n-3)-1/(2n-1)+1/(2n-1)-1/(2n+1)]
=(1/2)[1-1/(2n+1)]
=n/(2n+1)

注意 1/((2n)^2 - 1)
= 1/((2n+1)(2n-1))
= (1/(2n-1) - 1/(2n+1)) / 2
所以 2Sn = 2/2^2-1+2/4^2-1+……+2/(2n)^2-1
= (1- 1/3) + (1/3 - 1/5) + (1/5 - 1/7) + ... + (1/(2n-1) - 1/(2n+1))
= 1 - 1/(2n+1) = 2n/(2n+1)
得 Sn = n/(2n+1)

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