To find the sum to infinity of the series \(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\), we first need to recognize the type of series we are dealing with. This series is a geometric series. ### Step 1: Identify the first term and the common ratio In a geometric series, each term after the first is found by multiplying the previous term by a constant called the common ratio. - **First term (a)**: The first term of our series is \(1\). - **Common ratio (r)**: To find the common ratio, we can divide the second term by the first term: \[ r = \frac{\frac{1}{3}}{1} = \frac{1}{3} \] ### Step 2: Use the formula for the sum to infinity of a geometric series The formula for the sum to infinity \(S\) of a geometric series is given by: \[ S = \frac{a}{1 - r} \] where: - \(a\) is the first term, - \(r\) is the common ratio, and \(|r| < 1\). ### Step 3: Substitute the values into the formula Now we can substitute the values we identified into the formula: - \(a = 1\) - \(r = \frac{1}{3}\) Substituting these values into the formula gives: \[ S = \frac{1}{1 - \frac{1}{3}} = \frac{1}{\frac{2}{3}} = 1 \times \frac{3}{2} = \frac{3}{2} \] ### Conclusion: The sum to infinity of the series Thus, the sum to infinity of the series \(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots\) is: \[ \frac{3}{2} \] ### Answer: D. \( \frac{3}{2} \) ### Explanation of Other Options Now, let's analyze the other options to understand why they are incorrect: - **Option A: \( \frac{11}{3} \)**: This value is greater than \( \frac{3}{2} \) and does not fit the sum of a converging geometric series with the given terms. The sum of the series cannot exceed the first term divided by \(1 - r\). - **Option B: \( \frac{10}{3} \)**: Similar to option A, this value is also greater than \( \frac{3}{2} \) and does not represent the correct sum of the series. - **Option C: \( \frac{5}{2} \)**: This value is greater than \( \frac{3}{2} \) and does not align with the calculated sum. ### Common Pitfalls 1. **Misidentifying the series type**: It's crucial to recognize that this is a geometric series. If you mistakenly treat it as an arithmetic series, you will arrive at the wrong conclusion. 2. **Incorrectly calculating the common ratio**: Ensure that you divide the second term by the first term correctly to find the common ratio. 3. **Forgetting the condition for convergence**: The formula for the sum to infinity only applies when \(|r| < 1\). In this case, since \(r = \frac{1}{3}\), it converges. ### Revision Summary - The series is geometric with first term \(1\) and common ratio \(\frac{1}{3}\). - The sum to infinity formula is \(S = \frac{a}{1 - r}\). - Substituting \(a = 1\) and \(r = \frac{1}{3}\) gives \(S = \frac{3}{2}\). - The correct answer is \(D. \frac{3}{2}\).