Step 1 :We are given the geometric series \(\sum_{j=1}^{\infty} 4^{-2 j}\).
Step 2 :This is a geometric series with common ratio \(r = 4^{-2} = \frac{1}{16}\).
Step 3 :The sum of an infinite geometric series is given by the formula \(\frac{a}{1-r}\), where \(a\) is the first term of the series and \(r\) is the common ratio.
Step 4 :In this case, \(a = 4^{-2} = \frac{1}{16}\) and \(r = \frac{1}{16}\).
Step 5 :We can use this formula to find the sum of the series.
Step 6 :Substituting the values into the formula, we get \(\frac{0.0625}{1-0.0625} = 0.06666666666666667\).
Step 7 :Final Answer: The sum of the geometric series is \(\boxed{0.06666666666666667}\).