What is the sum of the series `1+(1)/(2)+(1)/(4)+(1)/(8)+`. . .?

Options

A. `1//2`
B. `3//2`
C. 2
D. `2//3`

Answer

To find the sum of the series \( S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \), we can recognize that this is a geometric series.

### Step-by-Step Solution:

1. **Identify the first term (A) and the common ratio (R)**:
- The first term \( A = 1 \).
- The second term is \( \frac{1}{2} \), and the first term is \( 1 \). Therefore, the common ratio \( R \) can be calculated as:
\[ R = \frac{A_2}{A_1} = \frac{\frac{1}{2}}{1} = \frac{1}{2} \]

2. **Check if the series is a geometric series**:
- A geometric series has a constant ratio between consecutive terms. Here, the ratio between any two consecutive terms is \( \frac{1}{2} \). Thus, this is indeed a geometric series.

3. **Use the formula for the sum of an infinite geometric series**:
- The formula for the sum \( S \) of an infinite geometric series is given by:
\[ S = \frac{A}{1 - R} \]
- Substituting the values of \( A \) and \( R \):
\[ S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]

4. **Conclusion**:
- Therefore, the sum of the series \( S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \) is \( 2 \).

### Final Answer:
The sum of the series is \( 2 \).