Question

[Lê Song Phương]

 Cho dãy số \(\left(x_n\right)^{+\infty}_{n=1}\) như sau: \(x_1=a>2\) và 

\(x_{n+1}=x_n^2-2,\forall n=1,2,...\)

 Tìm \(\lim\limits_{n\rightarrow+\infty}\left(\dfrac{1}{x_1}+\dfrac{1}{x_1x_2}+\dfrac{1}{x_1x_2x_3}+...+\dfrac{1}{x_1x_2...x_n}\right)\)

Answer 1

[Nguyễn Đức Trí]

\(x_1=a>2;x_{n+1}=x_n^2-2,\forall n=1,2,...\)

mà \(n\rightarrow+\infty\)

\(\Rightarrow a\rightarrow+\infty\Rightarrow x_n\rightarrow+\infty\)

\(\Rightarrow\lim\limits_{n\rightarrow+\infty}\dfrac{1}{x_n}=0\) \(\Rightarrow\lim\limits_{n\rightarrow+\infty}\left(\dfrac{1}{x_nx_{n+1}}\right)=0\)

\(\)\(\Rightarrow\lim\limits_{n\rightarrow+\infty}\left(\dfrac{1}{x_1}+\dfrac{1}{x_1x_2}+\dfrac{1}{x_1x_2x_3}+...+\dfrac{1}{x_1x_2...x_n}\right)=0\)

Answer 2

[Đặng Ninh Phượng]

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