MathDB

Let a_n=\frac{n^2+1}{\sqrt{n^4+4}}, n=1,2,3,\dots and let

b_n

be the product of

a_1,a_2,a_3,\dots ,a_n

. Prove that

\frac{b_n}{\sqrt{2}}=\frac{\sqrt{n^2+1}}{\sqrt{n^2+2n+2}},

and deduce that

\frac{1}{n^3+1}<\frac{b_n}{\sqrt{2}}-\frac{n}{n+1}<\frac{1}{n^3}

for all positive integers

n

Problem Statement