Determine the convergence of the given series: $\sum_{n=1}^{\infty} \frac{1}{2^{2 n+4}}$

Step 1 ：Recognize that this is a geometric series with common ratio \(\frac{1}{4}\), which is less than 1. Therefore, the series converges.

Step 2 ：Use the formula for the sum of an infinite geometric series, which is \(\frac{a}{1-r}\), where \(a\) is the first term and \(r\) is the common ratio.

Step 3 ：In this case, the first term \(a\) is \(\frac{1}{2^{2*1+4}} = \frac{1}{32}\) and the common ratio \(r\) is \(\frac{1}{4}\).

Step 4 ：Substitute \(a\) and \(r\) into the formula to get the sum of the series: \(\frac{\frac{1}{32}}{1-\frac{1}{4}} = \frac{1}{24}\).

Step 5 ：\(\boxed{\frac{1}{24}}\) is the sum of the series.