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B5

Part of 1963 Putnam

Problems(1)

Putnam 1963 B5

Source: Putnam 1963

5/1/2022
Let (an)(a_n ) be a sequence of real numbers satisfying the inequalities 0≤ak≤100an    for   n≤k≤2n    and    n=1,2,…, 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots, and such that the series ∑n=0∞an\sum_{n=0}^{\infty} a_n converges. Prove that lim⁡n→∞nan=0.\lim_{n\to \infty} n a_n = 0.
Putnaminequalitieslimitseries