Question
Find the value of \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots \infty } } } } when the layers of the square root go to infinite
Find the value of \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots \infty } } } } when the layers of the square root go to infinite
Let
when the layers of square root go to infinite
which makes no difference starting to count from the second square root.
Then, we get the following equation
Then,
Square both sides results in a quadratic equation
Using the root formula for a quadratic equation, we get
Cancel the negative solution, then,
In summary,
the value of \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\cdots \infty } } } } = \dfrac{1+\sqrt{13} }{2} when the layers of the square root go to infinite