求数列2²+1/2²-1,3²+1/3²-1,...的前n项和
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求数列2²+1/2²-1,3²+1/3²-1,...的前n项和
![求数列2²+1/2²-1,3²+1/3²-1,...的前n项和](/uploads/image/z/15646756-4-6.jpg?t=%E6%B1%82%E6%95%B0%E5%88%972%26%23178%3B%2B1%2F2%26%23178%3B-1%2C3%26%23178%3B%2B1%2F3%26%23178%3B-1%2C...%E7%9A%84%E5%89%8Dn%E9%A1%B9%E5%92%8C)
看通项an=[(n+1)²+1]/[(n+1)²-1]
=1+2/[(n+1)²-1]
=1+2/[(n+1-1)*(n+1+1)]
=1+2/[n(n+2)]
=1+1/n-1/(n+2)
∴ 前n项和=(1+1/1-1/3)+(1+1/2-1/4)+(1-1/3-1/5)+.+[1+1/n-1/(n+2)]
=n+1-1/3+1/2-1/4+1/3-1/5+.+1/(n-1)-1/(n+1)+1/n-1/(n+2)
=n+1+1/2-1/(n+1)-1/(n+2)
=1+2/[(n+1)²-1]
=1+2/[(n+1-1)*(n+1+1)]
=1+2/[n(n+2)]
=1+1/n-1/(n+2)
∴ 前n项和=(1+1/1-1/3)+(1+1/2-1/4)+(1-1/3-1/5)+.+[1+1/n-1/(n+2)]
=n+1-1/3+1/2-1/4+1/3-1/5+.+1/(n-1)-1/(n+1)+1/n-1/(n+2)
=n+1+1/2-1/(n+1)-1/(n+2)
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