Back to the Calculus 2

Calculus Level 4

Let $A= \lim_{n \to \infty } \int_{0}^{\frac \pi 2} \sqrt[n]{\sin^n x+ \cos^n x} \ dx$

Find $\underbrace{ A^{A^{A^{\cdots}}}}_{\text{ Indefinitely}}$

The answer is 2.

1 solution

Julian Poon
Nov 25, 2019

$\displaystyle N(x,y) = \lim_{n \rightarrow \infty} \sqrt[n]{x^n + y^n}$ is the $L^{\infty}$ norm and is equal to $\max(x,y)$ . Hence $\displaystyle A = 2 \int_{0}^{\frac{\pi}{4}} \cos x dx = \sqrt{2}$ . The power tower of $A$ is known to converge to $2$ .

Back to the Calculus 2

The answer is 2.

1 solution

0 pending reports