RMO District Round, Bucharest 2008, 11th Grade Let $m \geq 2 $ be a positive integer. Define the sequence $\left( x_n^{(m)} \right)_{n \geq 1} $ by : $x_n^{(m)} = \sum_{k=1}^n k^{-m} , \forall n \geq 1 $ a) prove that : $\left( x_n^{(m)} \right)_{n \geq 1} $ converges. b) Denote by $\ell_m $ the limit of $\left( x_n^{(m)} \right)_{n \geq 1} $. Determine the positive integers $ k $ for which there exists the nonzero and finite limit $\displaystyle\lim_{n \rightarrow \infty} n^k \left( \ell_m-x_n^{(m)} \right) $ |
Trích:
b,dùng bổ đề stolz kết quả $k= m-1 $ |
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