Home Competitions SEEMOUS 2014
SEEMOUS 2014
South Eastern European Mathematical Olympiad for University Students 2014.
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Problem 2
Consider the sequence ( x n ) (x_n) ( x n ) x 1 = 2 , x n + 1 = x n + 1 + x n 2 + 2 x n + 5 2 , n ≥ 2. x_1=2,\enspace x_{n+1}=\frac{x_n+1+\sqrt{x_n^2+2x_n+5}}2,\enspace n\ge2. x 1 = 2 , x n + 1 = 2 x n + 1 + x n 2 + 2 x n + 5 , n ≥ 2. y n = ∑ k = 1 n 1 x k 2 − 1 , n ≥ 1 y_n=\sum_{k=1}^n\frac1{x_k^2-1},\enspace n\ge1 y n = ∑ k = 1 n x k 2 − 1 1 , n ≥ 1
Problem 4
a) Prove that lim n → ∞ n ∫ 0 n arctan x n x ( x 2 + 1 ) d x = π 2 \lim_{n\to\infty}n\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx=\frac\pi2 lim n → ∞ n ∫ 0 n x ( x 2 + 1 ) arctan n x d x = 2 π lim n → ∞ n ( m ∫ 0 n arctan x n x ( x 2 + 1 ) d x − π 2 ) \lim_{n\to\infty}n\left(m\int^n_0\frac{\operatorname{arctan}\frac xn}{x(x^2+1)}dx-\frac\pi2\right) lim n → ∞ n ( m ∫ 0 n x ( x 2 + 1 ) arctan n x d x − 2 π )