The square root of 7 (√7) is an irrational number, meaning it cannot be expressed as a fraction or terminated decimal. It is approximately equal to 2.6457513111.


To evaluate √(7 + √7), we can substitute the value of √7 as 2.6457513111.
√(7 + √7) ≈ √(7 + 2.6457513111)
≈ √(9.6457513111)
≈ 3.1063350885


To evaluate √(7 + √(7 + √7)), we can substitute the value of √(7 + √7) as 3.1063350885.
√(7 + √(7 + √7)) ≈ √(7 + 3.1063350885)
≈ √(10.1063350885)
≈ 3.1784829807


As we continue to nest the square roots by adding the previous result, we can observe that the values are approaching a certain number. The process of repeatedly applying a function to its own output is known as iteration. In this case, we are iteratively applying the square root function.

The nested square root expressions converge to a value known as the limit. In this case, the limit is approximately 3.1784829807. This means that as we iterate the process of taking the square root and adding the previous result, the values will get closer and closer to 3.1784829807, but never reach it exactly.

This convergence can be proven mathematically using the concept of fixed points. A fixed point of a function is a value that remains unchanged when the function is applied. In this case, the square root of (7 + √(7 + √7)) is a fixed point of the square root function.

In conclusion, the nested square root expressions sqrt(7), sqrt(7 + sqrt(7)), and sqrt(7 + sqrt(7 + sqrt(7))) converge to a limit of approximately 3.1784829807.