Classic Limits Returns

Calculus Level 3

If x 1 = 3 x_1 = 3 and x n + 1 = 2 + x n x_{n+1} = \sqrt{2 + x_n} for n ≥ 1 n\geq1 , what is lim ⁡ n → ∞ x n \displaystyle \lim_{n\to\infty} x_n ?

− 1 -1 3 3 5 \sqrt5 2 2

Moulik Bhattacharya by Moulik Bhattacharya , 20, India

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1 solution

As n → ∞ n \rightarrow \infty we maw consider n + 1 → n n + 1 \rightarrow n . Then we may write x n + 1 → x n = y x_{n+1} \rightarrow x_{n} = y (if the limit exists, of course), which leads us to the following equation:

y = 2 + y . y = \sqrt{2 + y}.

Solving this equation results in the answear: 2. 2.

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