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

Options

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

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 is an infinite geometric series.

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

### Step 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:
\[ S = \frac{a}{1 - r} \]
where \( |r| < 1 \).

### Step 3: Substitute the values into the formula
Now, substituting the values of \( a \) and \( r \) into the formula:
\[ S = \frac{1}{1 - \left(-\frac{1}{2}\right)} \]

### Step 4: Simplify the expression
Now simplify the denominator:
\[ S = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]

### Conclusion
Thus, the sum of the series \( 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \ldots \) is:
\[ \boxed{\frac{2}{3}} \]