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sum a_k and sum(a_k)^(1-1/k)

Source: France 1989 P5

May 18, 2021
algebraSummation

Problem Statement

Let a1,a2,…,ana_1,a_2,\ldots,a_n be positive real numbers. Denote s=∑k=1nak and s′=∑k=1nak1−1k.s=\sum_{k=1}^na_k\text{ and }s'=\sum_{k=1}^na_k^{1-\frac1k}. (a) Let λ>1\lambda>1 be a real number. Show that s′<λs+λλ−1s'<\lambda s+\frac\lambda{\lambda-1}. (b) Deduce that s′<s+1\sqrt{s'}<\sqrt s+1.
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