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B1

Part of 1961 Putnam

Problems(1)

Putnam 1961 B1

Source: Putnam 1961

6/5/2022
Let a1,a2,a3,…a_1 , a_2 , a_3 ,\ldots be a sequence of positive real numbers, define sn=a1+a2+…+anns_n = \frac{a_1 +a_2 +\ldots+a_n }{n} and rn=a1−1+a2−1+…+an−1n.r_n = \frac{a_{1}^{-1} +a_{2}^{-1} +\ldots+a_{n}^{-1} }{n}. Given that lim⁡n→∞sn\lim_{n\to \infty} s_n and lim⁡n→∞rn\lim_{n\to \infty} r_n exist, prove that the product of these limits is not less than 1.1.
Putnamlimitinequalities