find the value of given domain and express in form of logarithm function `(1)/(2)-(1)/(2)*(1)/(2^(2))+(1)/(3)*(1)/(2^(3))-…….oo`=

Options

A. `log_(e )((2)/(3))`
B. `log_(e )((3)/(2))`
C. `log_(e )^(2)`
D. `log_(e )^(3)`

Answer

To solve the given series \( S = \frac{1}{2} - \frac{1}{2} \cdot \frac{1}{2^2} + \frac{1}{3} \cdot \frac{1}{2^3} - \ldots \) up to infinity, we can express it in a more manageable form and relate it to a logarithmic function.

### Step-by-Step Solution:

1. **Rewrite the Series**:
The series can be rewritten as:
\[ S = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \cdot \frac{1}{2^n} \]
This is a power series where the general term is \( \frac{(-1)^{n-1}}{n} \left( \frac{1}{2} \right)^n \).

2. **Recognize the Series**:
This series resembles the Taylor series expansion for \( \log(1+x) \):
\[ \log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \]
Here, we can identify \( x = -\frac{1}{2} \).

3. **Apply the Logarithmic Identity**:
Substituting \( x = -\frac{1}{2} \) into the logarithmic series gives:
\[ \log(1 - \frac{1}{2}) = -\frac{1}{2} - \frac{(-\frac{1}{2})^2}{2} + \frac{(-\frac{1}{2})^3}{3} - \ldots \]
This simplifies to:
\[ \log\left(\frac{1}{2}\right) = -\log(2) \]

4. **Relate to the Original Series**:
Since our series \( S \) is equal to \( -\log(1 + \frac{1}{2}) \), we have:
\[ S = -\log\left(\frac{3}{2}\right) \]
Therefore, we can express \( S \) as:
\[ S = \log\left(\frac{2}{3}\right) \]

5. **Final Result**:
The value of the series is:
\[ S = \log\left(\frac{3}{2}\right) \]

### Conclusion:
Thus, the value of the given series expressed in the form of a logarithm function is:
\[ \log\left(\frac{3}{2}\right) \]