Find `1+1/2+1/4+1/8+`…. Upto `oo`.

Options

A. `5 m//s`
B. `5sqrt(3) m//s`
C. `5/2sqrt(3) m//s`
D. `5/(4sqrt(3)) m//s`

Explanation

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

### Step-by-Step Solution:

1. **Identify the first term and common ratio**:
- The first term \( a \) of the series is \( 1 \).
- The common ratio \( r \) can be found by dividing the second term by the first term:
\[ r = \frac{\frac{1}{2}}{1} = \frac{1}{2} \]

2. **Check the condition for convergence**:
- For a geometric series to converge, the absolute value of the common ratio must be less than 1:
\[ |r| < 1 \quad \text{(Here, } r = \frac{1}{2} \text{, which satisfies this condition)} \]

3. **Use the formula for the sum of an infinite geometric series**:
- The sum \( S \) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a}{1 - r} \]
- Substituting the values of \( a \) and \( r \):
\[ S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]

4. **Final result**:
- Therefore, the sum of the series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots \) up to infinity is:
\[ S = 2 \]