Question

关于数列的问题

Answer 1

整理化简an=n/(2n-1)+n/(2n+1)
所以Sn=1+1/3+2/3+2/5+3/5+3/7+4/7+5/9.....n/(2n-1)+n/(2n+1)
注意规律除了1和n/(2n+1) 中间的都组合成了n-1个1了
所以Sn=1+(n-1)+n/(2n+1)=n+n/(2n+1)

Answer 2

an=4n^2/(4n^2-1)
=1+1/(4n^2-1)
=1+1/2*[1/(2n-1)-1/(2n+1)]

2an=2+1/(2n-1)-1/(2n+1)

2Sn=2a1+2a2+2a3+……+2an
＝2n+[1-1/3+1/3-1/5+1/5-1/7+……+1/(2n-1)-1/(2n+1)]
＝2n+1-1/(2n+1)
＝2n+2n/(2n+1)
Sn=n+n/(2n+1)