作业帮数列求和Sn=1/1*3+4/3*5+9/5*7+……n^2/(2n-1)(2n+1)
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数列求和Sn=1/1*3+4/3*5+9/5*7+……n^2/(2n-1)(2n+1)
n²/(2n-1)(2n+1)
=(n²-1/4+1/4)/(2n-1)(2n+1)
=(n²-1/4)/(2n-1)(2n+1)+(1/4)/(2n-1)(2n+1)
=(n+1/2)(n-/2)/(2n-1)(2n+1)+(1/8)[2/(2n-1)(2n+1)]
=(n+1/2)(n-/2)/[4(n+1/2)(n-/2)]+(1/8)[(2n+1)-(2n-1)]/(2n-1)(2n+1)]
=1/4+(1/8)[(2n+1)/(2n-1)(2n+1)-(2n-1)/(2n-1)(2n+1)]
=1/4+(1/8)[1/(2n-1)-1/(2n+1)]
所以原式=1/4+1/4+……+1/4+(1/8)[1-1/3+1/3-1/5+……+1/(2n-1)-1/(2n+1)]
=n/4+(1/8)[1-1/(2n+1)]
=n/4+n/(8n+4)
=(n²+n)/(4n+2)
=(n²-1/4+1/4)/(2n-1)(2n+1)
=(n²-1/4)/(2n-1)(2n+1)+(1/4)/(2n-1)(2n+1)
=(n+1/2)(n-/2)/(2n-1)(2n+1)+(1/8)[2/(2n-1)(2n+1)]
=(n+1/2)(n-/2)/[4(n+1/2)(n-/2)]+(1/8)[(2n+1)-(2n-1)]/(2n-1)(2n+1)]
=1/4+(1/8)[(2n+1)/(2n-1)(2n+1)-(2n-1)/(2n-1)(2n+1)]
=1/4+(1/8)[1/(2n-1)-1/(2n+1)]
所以原式=1/4+1/4+……+1/4+(1/8)[1-1/3+1/3-1/5+……+1/(2n-1)-1/(2n+1)]
=n/4+(1/8)[1-1/(2n+1)]
=n/4+n/(8n+4)
=(n²+n)/(4n+2)
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