Calculus Level 5

$P=\frac{1}{2}\left(\frac{\sqrt{1}}{\sqrt{2}}+\frac{\sqrt{2}}{\sqrt{1}}\right)\times\frac{1}{2}\left(\frac{\sqrt{2}}{\sqrt{3}}+\frac{\sqrt{3}}{\sqrt{2}}\right)\times \frac{1}{2}\left(\frac{\sqrt{3}}{\sqrt{4}}+\frac{\sqrt{4}}{\sqrt{3}}\right)\times \cdots$

The infinite product $$P$$ can be written as $$P=a\pi^{b}$$ for rational numbers $$a$$ and $$b$$. Enter $$a\times b$$ as your answer.

×