The sum of the series `1+ (1)/(4) + (1)/(16) + (1)/(64) + ...........oo` is

Options

A. `(8)/(7)`
B. `(6)/(5)`
C. `(5)/(4)`
D. `(4)/(3)`

Explanation

To find the sum of the infinite series \( S = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \ldots \), 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 \( A \) of the series is \( 1 \).
- The common ratio \( r \) can be found by dividing the second term by the first term:
\[ r = \frac{\frac{1}{4}}{1} = \frac{1}{4} \]

2. **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:
\[ S = \frac{A}{1 - r} \]
- Here, \( A = 1 \) and \( r = \frac{1}{4} \).

3. **Substitute the values into the formula**:
- Plugging in the values we have:
\[ S = \frac{1}{1 - \frac{1}{4}} \]

4. **Simplify the denominator**:
- Calculate \( 1 - \frac{1}{4} \):
\[ 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]

5. **Calculate the sum**:
- Now substitute back into the formula:
\[ S = \frac{1}{\frac{3}{4}} = \frac{1 \times 4}{3} = \frac{4}{3} \]

Thus, the sum of the series is \( \frac{4}{3} \).

### Final Answer:
The sum of the series is \( \frac{4}{3} \).