2^2/(2^2-1) ,4^2/(4^2-1) ,6^2/(6^2-1) ,8^2/(8^2-1)……这个数列怎么求和
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2^2/(2^2-1) ,4^2/(4^2-1) ,6^2/(6^2-1) ,8^2/(8^2-1)……这个数列怎么求和?
![2^2/(2^2-1) ,4^2/(4^2-1) ,6^2/(6^2-1) ,8^2/(8^2-1)……这个数列怎么求和](/uploads/image/z/1230522-42-2.jpg?t=2%5E2%2F%282%5E2-1%29+%2C4%5E2%2F%284%5E2-1%29+%2C6%5E2%2F%286%5E2-1%29+%2C8%5E2%2F%288%5E2-1%29%E2%80%A6%E2%80%A6%E8%BF%99%E4%B8%AA%E6%95%B0%E5%88%97%E6%80%8E%E4%B9%88%E6%B1%82%E5%92%8C)
Sn=2^2/(2^2-1)+4^2/(4^2-1)+6^2/(6^2-1)+……+n^2/((2n)^2-1)
=[1+1/(2^2-1)]+[1+1/(4^2-1)]+[1+1/(6^2-1)]+……+[1+1/((2n)^2-1)]
=n+0.5×[2/(2-1)×(2+1)+2/(4-1)×(4+1)+2/(6-1)×(6+1)+……+2/(2n-1)×(2n+1)]
=n+0.5×[(3-1)/1×3+(5-3)/3×5+(7-5)/5×7+……+((2n+1)-(2n-1))/(2n-1)×(2n+1)]
=n+0.5×[(1/1-1/3)+(1/3-1/5)+(1/5-1/7)+……+(1/(2n-1)-1/(2n+1)]
=n+0.5×[1-1/(2n+1)]
=n+n/(2n+1)
=[1+1/(2^2-1)]+[1+1/(4^2-1)]+[1+1/(6^2-1)]+……+[1+1/((2n)^2-1)]
=n+0.5×[2/(2-1)×(2+1)+2/(4-1)×(4+1)+2/(6-1)×(6+1)+……+2/(2n-1)×(2n+1)]
=n+0.5×[(3-1)/1×3+(5-3)/3×5+(7-5)/5×7+……+((2n+1)-(2n-1))/(2n-1)×(2n+1)]
=n+0.5×[(1/1-1/3)+(1/3-1/5)+(1/5-1/7)+……+(1/(2n-1)-1/(2n+1)]
=n+0.5×[1-1/(2n+1)]
=n+n/(2n+1)