Find the limit of the following sequence or determine that the sequence diverges. \[ \left\{\left(1+\frac{4}{n}\right)^{5 n}\right\} \]

To find the limit of the sequence \[ a_n = \left(1+\frac{4}{n}\right)^{5n}, \] we can utilize the fact that this expression resembles the definition of the exponential function involving limits. Specifically, we can rewrite the term inside the parentheses as follows: \[ \left(1+\frac{4}{n}\right)^{5n} = \left(\left(1+\frac{4}{n}\right)^{n}\right)^{5}. \] As \( n \to \infty \), we recognize that \[ \left(1+\frac{4}{n}\right)^{n} \to e^{4}. \] This is because the expression \(\left(1+\frac{x}{n}\right)^{n}\) approaches \(e^{x}\) as \(n\) goes to infinity for any fixed \(x\). In our case, we have \(x = 4\). Thus, we have: \[ \left(1+\frac{4}{n}\right)^{n} \to e^{4} \quad \text{as } n \to \infty. \] Therefore, raising this to the power of 5 gives us: \[ \left(1+\frac{4}{n}\right)^{5n} = \left(\left(1+\frac{4}{n}\right)^{n}\right)^{5} \to (e^{4})^{5} = e^{20}. \] Thus, the limit of the sequence is \[ \lim_{n \to \infty} a_n = e^{20}. \] In conclusion, the limit of the sequence \( a_n = \left(1+\frac{4}{n}\right)^{5n} \) as \( n \to \infty \) is \[ \boxed{e^{20}}. \]