Nested stuff

Calculus Level 3

$\large y = \int_{-1}^{1} \left ( \sqrt{ 1 - \sqrt {1 - \sqrt{ 1 -\dots \sqrt{1 - x^2}}}} \right) \text{d}x \,$

Consider the nested function above, what is the value of $y$ in the limit of infinite nesting? Give your answer to 3 decimal places.

The answer is 1.236.

1 solution

Alex Li
Jun 4, 2015

As the number of square roots approaches infinity, the function becomes constant. To find out the value of the infinite nested square roots, let it be $c$ . We have $c=\sqrt{1-c}$ . Solving, $c=\frac{\sqrt{5}-1}{2}$ . Thus, the integral evaluates as $\frac{\sqrt{5}-1}{2}\times(1-(-1))=\boxed{\sqrt{5}-1}$ .

Moderator note:

Does the integrand converge? How?

Nested stuff

The answer is 1.236.

1 solution

Moderator note:

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