Does $\lim_{n \to \infty}\operatorname{length}\{ x^n(1-x)^n:x\in[0,1],n\in \Bbb R^+\}=1?$ I'd like to prove this result analytically, but I'm not sure how.
The formula for arc length that I've been using is $s=\int_0^1\sqrt{1+(dy/dx)^2}dx.$
2026-05-16 21:04:37.1778965477
Prove that the limit equals $1$
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Write down $dy/dx$ and use the inequality $x(1-x) \leq \frac 1 4$. You will see that $s \leq \sqrt {1+\frac {n^{2}} {4^{n-1}}}$. From this it is clear that $s \to 1$ ( $s \geq 1$ is obvious). [$x(1-x)$ attains its maximum at $x=\frac 1 2$].