MathNet 2026 — Part 05 Question 125

Suppose $a_1 = 3$, $a_{n+1} = \frac{1}{2}(a_n^2 + 1)$, $n = 1, 2, \dots$. Prove that
$$
\sum_{k=1}^{n} \frac{1}{1+a_k} \lt \frac{1}{2}
$$