Factorials at infinity

Calculus Level 4

\[ \lim_{n \to \infty} \left [ { { \left( \begin{matrix} 2n \\ n \end{matrix} \right) }^{ 1/n } }+ { { \left( \frac { (3n)! }{ { (n!) }^{ 3 } } \right) }^{ 1/n } } \right ] = \ ? \]

Bonus: Find the general form for \( \displaystyle { { \left( \frac { (m n)! }{ { (n!) }^{ m } } \right) }^{ 1/n } } \) for positive integer \(m\).

Details and Assumptions:

  • \( { { \left( \begin{matrix} a \\ b \end{matrix} \right) }} \) denote the binomial coefficient.
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