Question 65
Prove that, for every positive integer $n$, there exists a unique $c_n \in (0, 1)$ such that
$$
\int_0^1 \frac{1}{1+x^n} dx = \frac{1}{1+(c_n)^n},
$$
and evaluate the limit $\lim_{n \to \infty} n(c_n)^n$.
$$
\int_0^1 \frac{1}{1+x^n} dx = \frac{1}{1+(c_n)^n},
$$
and evaluate the limit $\lim_{n \to \infty} n(c_n)^n$.