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Câu 3 trang 130 SGK Đại số và Giải tích 11 Nâng cao

Chứng minh rằng
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Chứng minh rằng các dãy số (un) sau đây có giới hạn 0 :

LG a

 \({u_n} = {\left( {0,99} \right)^n}\)

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: \(\left| {0,99} \right| < 1\) nên \(\lim {u_n} = \lim {\left( {0,99} \right)^n} = 0\)

LG b

\({u_n} = {{{{\left( { - 1} \right)}^n}} \over {{2^n} + 1}}\)

Lời giải chi tiết:

\(\eqalign{
& \left| {{u_n}} \right| = \left| {{{{{\left( { - 1} \right)}^n}} \over {{2^n} + 1}}} \right| = {1 \over {{2^n} + 1}}\cr & <\frac{1}{{{2^n}}} =  {\left( {{1 \over 2}} \right)^n}\cr &\lim {\left( {{1 \over 2}} \right)^n} = 0 \cr 
& \Rightarrow \lim {u_n} = 0 \cr} \)

LG c

\({u_n} = - {{\sin {{n\pi } \over 5}} \over {{{\left( {1,01} \right)}^n}}}\)

Lời giải chi tiết:

\(\eqalign{
༺& \left| {{u_n}} \right| = {{\left| {\sin {{n\pi } \over 5}} \right|} \over {{{\left( {1,01} \right)}^n}}} \le \frac{1}{{1,{{01}^n}}} = {\left( {{1 \over {1,01}}} \right)^n},\cr &\lim {\left( {{1 \over {1,01}}} \right)^n} = 0 \Rightarrow \lim {u_n} = 0 \cr} \)

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