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 it resembles the expression for the number \( e \). As \( n \) approaches infinity, we can express the limit in terms of \( e \): \[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(1 + \frac{4}{n}\right)^{5n} \] This can be rewritten in a useful form: \[ = \lim_{n \to \infty} \left(\left(1 + \frac{4}{n}\right)^{n}\right)^{5} \] Now, we know that: \[ \lim_{n \to \infty} \left(1 + \frac{4}{n}\right)^{n} = e^4 \] So, substituting this back into our limit: \[ = (e^4)^5 = e^{20} \] Thus, the limit of the given sequence is \[ \boxed{e^{20}}. \]