Solution to problem 1110

Let {a_{n}}_{n≥1} be a monotone increasing sequence of positive numbers with lim_{n→∞} a_{n} = l ≥ 0. Prove that

∑_{k=1}^{∞}(a_{2k−1} + a_{2k})\frac{a_{2k+1} − a_{2k}}{a_{2k+1} + a_{2k}} ≤ l − \frac{a_{1} + a_{2}}{2}

Proof Since

0 < \frac{a_{2k−1} + a_{2k}}{a_{2k+1} + a_{2k}} ≤ 1

we have

∑_{k=1}^{n}(a_{2k−1} + a_{2k})\frac{a_{2k+1} − a_{2k}}{a_{2k+1} + a_{2k}} ≤ ∑_{k=1}^{n}(a_{2k+1} − a_{2k}) \underbrace{=}_{telescoping} a_{2n+1} − a_{2}

The limit n → ∞ yields

∑_{k=1}^{∞}(a_{2k−1} + a_{2k})\frac{a_{2k+1} − a_{2k}}{a_{2k+1} + a_{2k}} \underbrace{≤}_{a_{k}≤l ∀ k} l − a_{2} ≤ l − \frac{a_{1} + a_{2}}{2}

since l − a_{2} ≤ l − \frac{a_{1} + a_{2}}{2} due to the monotonicity increasing of \{a_{n}\}.