an=2n
1/an=1/(2n)
1/an^2=1/(2n)^2
1/a1^2+1/a2^2+1/a3^2+...........+1/an^2
=1/(4*1^2)+1/(4*2^2)+1/(4*3^2)+.........+1/(4*n^2)
=1/4*(1/1^2+1/2^2+1/3^2+..........+1/n^2)
<1/4*[1/1^2+1/(1*2)+1/(2*3)+.........+1/n*(n+1)]
=1/4*[1+1-1/2+1/2-1/3+........+1/n-1/(n+1)]
=1/4*(2-1/(n+1)]
=1/2-1/4*1/(n+1)
即1/a1^2+1/a2^2+1/a3^2+...........+1/an^2<1/2
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