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Problems(1)

Source: Osaka Institute of Tecnology entrance exam 2006

\{a_{n}\}

a_{1}=\frac{\pi}{4},\ a_{n}=\int_{0}^{\frac{1}{2}}(\cos \pi x+a_{n-1})\cos \pi x\ dx\ \ (n=2,3,\cdots)

\lim_{n\to\infty}a_{n}