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Câu 4 trang 130 SGK Đại số và Giải tích 11 Nâng cao

Cho dãy số (un)
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 Cho dãy số (un) với  \({u_n} = {n \over {{3^n}}}\)

LG a

Chứng minh rằng \({{{u_{n + 1}}} \over {{u_n}}} \le {2 \over 3}\) với mọi n.

Lời giải chi tiết:

Ta có:

\(\eqalign{
& {{{u_{n + 1}}} \over {{u_n}}} = {{n + 1} \over {{3^{n + 1}}}}:{n \over {{3^n}}}  = \frac{{n + 1}}{{{{3.3}^n}}}.\frac{{{3^n}}}{n}\cr &= {1 \over 3}.{{n + 1} \over n} = {1 \over 3}\left( {1 + {1 \over n}} \right) \cr 
🐭&  \le {1 \over 3}(1+1)={2 \over 3},\forall n \ge 1. \cr} \)

(Vì \(\forall n \ge 1 \Rightarrow \dfrac{1}{n} \le 1\))

LG b

Bằng phương pháp qui nạp, chứng minh rằng \(0 < {u_n} \le {\left( {{2 \over 3}} \right)^n}\) với mọi n.

Lời giải chi tiết:

Rõ ràng \(u_n> 0, ∀n ≥ 1\). Ta chứng minh  \({u_n} \le {\left( {{2 \over 3}} \right)^n}\,\,\,\,\left( 1 \right)\) +) Với \(n = 1\) ta có  \({u_1} = {1 \over 3} \le {2 \over 3}\) Vậy (1) đúng với \(n = 1\) +) Giả sử (1) đúng với \(n = k\), tức là ta có: \({u_k} \le {\left( {{2 \over 3}} \right)^k}\) Khi đó \(\frac{{{u_{k + 1}}}}{{{u_k}}} \le \frac{2}{3} \Leftrightarrow {u_{k + 1}} \le {2 \over 3}{u_k}\) (theo câu a) \( \Rightarrow {u_{k + 1}} \le {2 \over 3}.{\left( {{2 \over 3}} \right)^k} = {\left( {{2 \over 3}} \right)^{k + 1}}\) Vậy (1) đúng với \(n = k + 1\) nên (1) đúng với mọi \(n\).

LG c

Phương pháp giải:

Sử dụng các định lý: +) Cho hai dãy số \(\left( {{u_n}} \right),\left( {{v_n}} \right)\). Nếu \(\left| {{u_n}} \right| \le {v_n}\) với mọi n và \(\lim {v_n} = 0\) thì \(\lim {u_n} = 0\). +) Nếu \(\left| q \right| < 1\) thì \(\lim {q^n} = 0\).

Lời giải chi tiết:

Ta có: \(0 < {u_n} \le {\left( {{2 \over 3}} \right)^n} \Rightarrow \left| {{u_n}} \right| \le {\left( {{2 \over 3}} \right)^n}\) Mà  \(\lim {\left( {{2 \over 3}} \right)^n} = 0\) \( \Rightarrow \lim \left| {{u_n}} \right| = 0 \Rightarrow \lim {u_n} = 0\)

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