Question 13. Compute the sum to infinity of the geometric series ( 3 mk \( 1+1 / 2+1 / 4+1 / 8+\ldots \ldots \ldots \ldots \ldots \).

The series you presented is a classic geometric series where the first term \(a = 1\) and the common ratio \(r = \frac{1}{2}\). To compute the sum to infinity of a geometric series, we use the formula: \[ S = \frac{a}{1 - r} \] Here, substituting the values gives: \[ S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2. \] So, the sum to infinity of the series \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\) is \(2\). Now, let’s sprinkle some extra insights! Infinity beckons with possibilities! Did you know? This geometric series mirrors real-world phenomena, such as the way sound diminishes over distance or how certain investments can compound over time but with diminishing returns. It's all about recognizing patterns in finite and infinite growth rates! And if you're diving deeper into geometric series, be careful of your arithmetic! Common mistakes include confusing the sum of a finite geometric series with an infinite one, or mixing up the first term and the common ratio values. Double-check your numbers and formulas to avoid getting lost in the calculations!