【Math】2. Infinite Product、A Constrained Polynomial

记录算法题解

124人浏览 · 2020-12-27 10:42:47

记录算法题解 · 2020-12-27 10:42:47 发布

🔢 Math

Problem 1 (Easy–Medium): Infinite Product

Evaluate

$∏n=2∞n3−1n3+1\prod_{n=2}^{\infty} \frac{n^3 - 1}{n^3 + 1}$

Hint: Factor both numerator and denominator using $a3±1a^3 \pm 1$ , then look for telescoping patterns.

Solution

Let $ω=ei2π3\omega=e^{i\frac{2\pi}{3}}$ , then $n3−1=(n−1)(n−ω)(n+1+ω),n3+1=(n+1)(n+ω)(n−1−ω)n^3-1=(n-1)(n-\omega)(n+1+\omega), n^3+1=(n+1)(n+\omega)(n-1-\omega)$ , so $∏n=2∞n3−1n3+1=∏n=2∞(n−1)(n−ω)(n+1+ω)(n+1)(n+ω)(n−1−ω)=2×12+ω×11−ω=23\prod_{n=2}^{\infty} \frac{n^3 - 1}{n^3 + 1}=\prod_{n=2}^{\infty} \frac{(n-1)(n-\omega)(n+1+\omega)}{(n + 1)(n+\omega)(n-1-\omega)}=2\times \frac{1}{2+\omega}\times \frac{1}{1-\omega}=\frac{2}{3}$

Problem 2 (Medium–Hard): A Constrained Polynomial

Let $p (x)$ be a monic polynomial of degree $4$ satisfying $p (1) = 1$ , $p (2) = 2$ , and $p (3) = 3$ . Note that these three constraints do not uniquely determine $p$ — there is a one-parameter family of solutions.

Show that the value of $p (0) + p (4)$ is nonetheless the same for every such polynomial, and find it.

Hint: Consider $q (x) = p (x) - x$ . What do you know about its roots?

Solution

Consider $f (x) = p (x) - x$ , it has roots $1, 2, 3$ thus $f(x)=(x-1)(x-2)(x-3)(x-x_0)$ . $f(0)=6x_0, f(4)=6(4-x_0)$ , thus $f (0) + f (4) = p (0) + p (4) - 4 = 24$ , $p (0) + p (4) = 28$ .