To solve the common problem involving the infinite nested square root of the form ( \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}} ), follow these steps:

Step 1: Define the limit

Let ( L = \sqrt{2 + \sqrt{2 + \sqrt{2 + \dots}}} ).

Since the nested structure repeats infinitely, the expression inside the outermost square root is ( 2 + L ). Thus:
[ L = \sqrt{2 + L} ]

Step 2: Solve for ( L )

Square both sides to eliminate the square root:
[ L^2 = 2 + L ]

Rearrange into a quadratic equation:
[ L^2 - L - 2 = 0 ]

Factor the quadratic:
[ (L - 2)(L + 1) = 0 ]

Step 3: Select the valid solution

Since ( L ) is a positive number (square root of a positive value), we discard ( L = -1 ).

Thus, ( L = 2 ).

Answer: (\boxed{2})

作者声明:本文包含人工智能生成内容。