MathDB
IMC 1996 Problem 12

Source: IMC 1996

March 4, 2021
inequalitiesreal analysis

Problem Statement

i) Prove that for every sequence (an)n∈N(a_{n})_{n\in \mathbb{N}}, such that an>0a_{n}>0 for all n∈Nn \in \mathbb{N} and ∑n=1∞an<∞\sum_{n=1}^{\infty}a_{n}<\infty, we have ∑n=1∞(a1a2⋯an)1n<e∑n=1∞an.\sum_{n=1}^{\infty}(a_{1}a_{2} \cdots a_{n})^{\frac{1}{n}}< e\sum_{n=1}^{\infty}a_{n}. ii) Prove that for every ϵ>0\epsilon>0 there exists a sequence (bn)n∈N(b_{n})_{n\in \mathbb{N}} such that bn>0b_{n}>0 for all n∈Nn \in \mathbb{N} and ∑n=1∞bn<∞\sum_{n=1}^{\infty}b_{n}<\infty and ∑n=1∞(b1b2⋯bn)1n>(e−ϵ)∑n=1∞bn.\sum_{n=1}^{\infty}(b_{1}b_{2} \cdots b_{n})^{\frac{1}{n}}> (e-\epsilon)\sum_{n=1}^{\infty}b_{n}.
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