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4

Part of 1996 IMC

Problems(1)

IMC 1996 Problem 4

Source: IMC 1996

3/5/2021
Let a1=1a_{1}=1, an=1n∑k=1n−1akan−ka_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k} for n≥2n\geq 2. Show that i) lim sup⁡n→∞∣an∣1n<2−12\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}; ii) lim sup⁡n→∞∣an∣1n≥23\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}
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