RMO District Round, Bucharest 2008, 11th Grade

[thaithuan_GC]

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) $
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[Quân -k47DHV]

Nguyên văn bởi thaithuan_GC

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) $

a,nhận xét $(x_{n}) $tăng bị chặn bởi 2 .nên có giới hạn
b,dùng bổ đề stolz kết quả $k= m-1 $
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