What is the sum of the infinite series `1-(1)/(3)+(1)/(9)-(1)/(27)+…` ?

Options

A. `(2)/(3)`
B. `(3)/(4)`
C. 1
D. `(4)/(3)`

Explanation

To find the sum of the infinite series \(1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} + \ldots\), we can recognize that this is a geometric series.

### Step-by-Step Solution:

1. **Identify the first term and the common ratio**:
- The first term \(a_1\) of the series is \(1\).
- The second term \(a_2\) is \(-\frac{1}{3}\).
- To find the common ratio \(r\), we can use the formula:
\[ r = \frac{a_2}{a_1} = \frac{-\frac{1}{3}}{1} = -\frac{1}{3} \]

2. **Write the formula for the sum of an infinite geometric series**:
- The sum \(S\) of an infinite geometric series can be calculated using the formula:
\[ S = \frac{a_1}{1 - r} \]
- Here, \(a_1 = 1\) and \(r = -\frac{1}{3}\).

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

4. **Simplify the expression**:
- The denominator can be simplified:
\[ 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]
- Therefore, we have:
\[ S = \frac{1}{\frac{4}{3}} = \frac{1 \times 3}{4} = \frac{3}{4} \]

5. **Conclusion**:
- The sum of the infinite series is:
\[ S = \frac{3}{4} \]