<s>[INST] Problem with sequence and rational function Let $a_n=\frac {(2n+1)(n^2-2n+2)}{(n^2+1)(n^4+4)}$, for every $n \ge 1$. Show that
$$(1+a_1)(1+a_2)\dots(1+a_n) \lt \sqrt e.$$
MY TRY: Let us denote $x_n=\ln(1+a_1)+\ln (1+a_2) + \dots + \ln(1+a_n)$. The problem is now to show that $x_n \lt \frac 1 2.$ I showed that $x_n$ is increasing, so I think about proving that $\lim_{n \to \infty} x_n= \frac 1 2$. I am stuck now.
 [/INST] $$n^4 + 4 = (n^2 + 2)^2 - (2n)^2 = (n^2 + 2n + 2)(n^2 - 2n + 2)$$
Thus $$a_n = \frac{2n+1}{(n^2 + 1)(n^2 + 2n + 2)}$$
Notice that the difference of the terms in the denominator is the numerator.
Thus if $b_n = \frac{1}{n^2 + 1}$ then
$$a_n = b_n - b_{n+1}$$
Note that $b_1 = \frac{1}{2}$.
Now use $\log (1 + a) \lt a$ to get
$$\sum \log(1+a_k) \le \sum a_k = b_{1} - b_{n+1} \lt b_1 = \frac{1}{2}$$
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