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Explanation & Answer
n=1
left = 1/2 < 1/sqrt(2)
assume the inequality hold for n=k, denote the left side as s(k), then we have s(k)<1/sqrt(k+1)
n=k+1, the left side
s(k)*(2k+1)/(2k+2)<1/sqrt(k+1) *(2k+1)/(2k+2) = 1/sqrt(k+1) *(2k+2)/(2k+3)
=2sqrt(k+1)/(2k+3)
=2sqrt(k+1)/(k+1+k+2) = 2/(sqrt(k+1)+(k+2)/sqrt(k+1))< 2/{2sqrt(sqrt(k+1)*k+2)/sqrt(k+1))}
= 1/sqrt(k+2)
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