\(u_1=1=\sqrt{2.0+1}\)
\(u_2=\sqrt{2+u_1^2}=\sqrt{3}=\sqrt{2.1+1}\)
\(u_3=\sqrt{2+u_2^2}=\sqrt{5}=\sqrt{2.2+1}\)
\(\Rightarrow u_n=\sqrt{2\left(n-1\right)+1}=\sqrt{2n-1}\)
\(\Rightarrow v_n=\sqrt{\dfrac{2n-1}{n}}=\sqrt{2-\dfrac{1}{n}}\)
\(\Rightarrow\lim\left(v_n\right)=\lim\sqrt{2-\dfrac{1}{n}}=\sqrt{2}\)
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