已知数列{an}中,an=1/n(n+2).求n趋向正无穷大时Sn的极限?

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已知数列{an}中,an=1/n(n+2).求n趋向正无穷大时Sn的极限?
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已知数列{an}中,an=1/n(n+2).求n趋向正无穷大时Sn的极限?
已知数列{an}中,an=1/n(n+2).求n趋向正无穷大时Sn的极限?

已知数列{an}中,an=1/n(n+2).求n趋向正无穷大时Sn的极限?
an=1/n(n+2)=1/2*(1/n-1/(n+2))
Sn=a1+a2+.+an
=1/2*(1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+.+1/(n-2)-1/n+1/(n-1)-1/(n+1)+1/n-1/(n+2))
=1/2(1+1/2-1/(n+1)-1/(n+2))
=1/2(3/2-(2n+3)/(n^2+3n+2)
=(3n^2+5n)/2(n^2+3n+2)
=(3+5/n)/2(1+3/n+2/n^2)
所以limn->∞Sn=limn->∞(3+5/n)/2(1+3/n+2/n^2)=(3+0)/2(1+0+0)=3/2