$\alpha,\ \beta$を \[ \alpha=\lim_{n \to \infty} \left( \frac{(3n+1)(3n+2)(3n+3) \cdots (3n+n)}{(n+1)(n+2)(n+3) \cdots (n+n)} \right)^{\frac{1}{n}} \] および \[ \beta=\lim_{n \to \infty} \left( \frac{(3n^2+1^2)(3n^2+2^2)(3n^2+3^2) \cdots (3n^2+n^2)}{(n^2+1^2)(n^2+2^2)(n^2+3^2) \cdots (n^2+n^2)} \right)^{\frac{1}{n}} \] とおく．このとき$\alpha<\beta$を示せ．また，$\alpha$と$\beta$の値をそれぞれ求めよ．