Playing with Radicals

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.

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