To find the sum to infinity of the series \( \frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \ldots \), we first need to identify the pattern in the series and determine if it is a geometric series. ### Step 1: Identify the Series Type A geometric series is defined as a series where each term after the first is found by multiplying the previous term by a constant ratio \( r \). Let's examine the terms given: - First term \( a_1 = \frac{1}{2} \) - Second term \( a_2 = \frac{1}{6} \) - Third term \( a_3 = \frac{1}{18} \) Now, let's find the ratio \( r \) between consecutive terms: \[ r = \frac{a_2}{a_1} = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{6} \times \frac{2}{1} = \frac{2}{6} = \frac{1}{3} \] Next, we check the ratio between the second and third terms: \[ r = \frac{a_3}{a_2} = \frac{\frac{1}{18}}{\frac{1}{6}} = \frac{1}{18} \times \frac{6}{1} = \frac{6}{18} = \frac{1}{3} \] Since the ratio \( r \) is consistent, we confirm that this is a geometric series with: - First term \( a = \frac{1}{2} \) - Common ratio \( r = \frac{1}{3} \) ### Step 2: Sum to Infinity Formula The formula for the sum to infinity \( S \) of a geometric series is given by: \[ S = \frac{a}{1 - r} \] This formula is valid only if the absolute value of the common ratio \( |r| < 1 \). In our case, \( r = \frac{1}{3} \), which satisfies this condition. ### Step 3: Calculate the Sum to Infinity Now, we can substitute the values of \( a \) and \( r \) into the formula: \[ S = \frac{\frac{1}{2}}{1 - \frac{1}{3}} = \frac{\frac{1}{2}}{\frac{2}{3}} \] To simplify this, we multiply by the reciprocal of the denominator: \[ S = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4} \] ### Conclusion: Correct Option Thus, the sum to infinity of the series is: \[ \boxed{\frac{3}{4}} \] ### Step 4: Explanation of Other Options - **Option A: \( \frac{2}{3} \)**: This value does not match our calculated sum. It may arise from a miscalculation or misunderstanding of the series' terms. - **Option B: \( \frac{1}{3} \)**: This is also incorrect. It could be a result of incorrectly applying the sum formula or misidentifying the series. - **Option D: \( 1 \)**: This option is incorrect as well. It suggests a misunderstanding of the convergence of the series, as the sum is less than 1. ### Revision Summary - The series \( \frac{1}{2}, \frac{1}{6}, \frac{1}{18}, \ldots \) is a geometric series with first term \( a = \frac{1}{2} \) and common ratio \( r = \frac{1}{3} \). - The sum to infinity of a geometric series is calculated using \( S = \frac{a}{1 - r} \). - The correct sum to infinity for this series is \( \frac{3}{4} \). - Always check that \( |r| < 1 \) to ensure the series converges before applying the sum formula.