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Bài 2.5 trang 49 SGK Toán 11 tập 1 - Cùng khám phá

Chứng minh rằng các dãy số (un) cho bởi các công thức sau đây bị chặn:
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Đề bài

Chứng minh rằng các dãy số (un) cho bởi các công thức sau đây bị chặn:

a) \({u_n} = 2 + \frac{1}{n};\) b) \({u_n} = \frac{1}{{n\left( {n + 1} \right)}};\) c) \({u_n} = \sin \left( n \right) + \cos \left( n \right).\)

Phương pháp giải - Xem chi tiết

Dãy số \(\left( {{u_n}} \right)\) bị chặn khi \(m \le {u_n} \le M\forall n\) nguyên dương.

Lời giải chi tiết

a)\(\begin{array}{l}{u_n} = 2 + \frac{1}{n}\\n \ge 1 \Leftrightarrow 0 < \frac{1}{n} \le 1 \Leftrightarrow 2 < 2 + \frac{1}{n} \le 3\end{array}\)Vậy dãy số đã cho là dãy bị chặn.b)\(\begin{array}{l}{u_n} = \frac{1}{{n\left( {n + 1} \right)}}\\n \ge 1 \Leftrightarrow n + 1 \ge 2 \Rightarrow n\left( {n + 1} \right) \ge 2\\ \Rightarrow 0 < \frac{1}{{n\left( {n + 1} \right)}} \le \frac{1}{2}\end{array}\)Vậy dãy số đã cho là dãy bị chặn.c)\(\begin{array}{l}{u_n} = \sin \left( n \right) + \cos \left( n \right)\\\left\{ \begin{array}{l} - 1 \le \sin \left( n \right) \le 1\\ - 1 \le \cos \left( n \right) \le 1\end{array} \right.\\ \Rightarrow  - 2 \le \sin \left( n \right) + \cos \left( n \right) \le 2\end{array}\)Vậy dãy số đã cho là dãy bị chặn.

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