Question

设S1=1+1/(1^2)+1/(2^2),S2=1+1/(2^2)+1/(3^2),S3=1+1/(3^2)+1/(4^2)......Sn=1+1/[n^2+1/(n+1)^2].

设S1=1+1/(1^2)+1/(2^2),S2=1+1/(2^2)+1/(3^2),S3=1+1/(3^2)+1/(4^2)......Sn=1+1/[n^2+1/(n+1)^2].设S=√S1+√S2+√S3+......+√Sn，则S=? (用含n的代数式表示，其中n为正整数）

Answer 1

√S1=1+1/(1×2) √S2=1+1/(2×3) ….√Sn=1+1/(n×(n+1))
S=(1+1+…..+1)+1/(1×2)+1/(2×3)+…+1/(n×(n+1))=n+[1-1/(n+1)]
= n+n/(n+1)
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Answer 2

√S1=1+1/(1×2) √S2=1+1/(2×3) ….√Sn=1+1/(n×(n+1))
S=(1+1+…..+1)+1/(1×2)+1/(2×3)+…+1/(n×(n+1))
=n+{(1-1/2)+(1/2-1/3)+(1/3-1/4)+......+[1/n-1/(n+1)]}
=n+[1-1/2+1/2-1/3+1/3-1/4+......+1/n-1/(n+1)]
=n+[1-1/(n+1)]
= n+n/(n+1)