`(4)/(1.3)-(6)/(2.4)+(12)/(5.7)-(14)/(6.8)+….=`

Options

A. `log_(e )((2)/(e ))`
B. `log_(e )((e )/(2))`
C. `log_(e)2`
D. 0

Answer

To solve the expression

\[ \frac{4}{1 \cdot 3} - \frac{6}{2 \cdot 4} + \frac{12}{5 \cdot 7} - \frac{14}{6 \cdot 8} + \ldots \]

we will first rewrite the numerators in a more manageable form.

### Step 1: Rewrite the Numerators
We can express the numerators as follows:
- \(4 = 1 + 3\)
- \(6 = 2 + 4\)
- \(12 = 5 + 7\)
- \(14 = 6 + 8\)

Thus, we can rewrite the terms:
\[ \frac{4}{1 \cdot 3} = \frac{1 + 3}{1 \cdot 3} = \frac{1}{1 \cdot 3} + \frac{3}{1 \cdot 3} \]
\[ \frac{6}{2 \cdot 4} = \frac{2 + 4}{2 \cdot 4} = \frac{2}{2 \cdot 4} + \frac{4}{2 \cdot 4} \]
\[ \frac{12}{5 \cdot 7} = \frac{5 + 7}{5 \cdot 7} = \frac{5}{5 \cdot 7} + \frac{7}{5 \cdot 7} \]
\[ \frac{14}{6 \cdot 8} = \frac{6 + 8}{6 \cdot 8} = \frac{6}{6 \cdot 8} + \frac{8}{6 \cdot 8} \]

### Step 2: Separate the Terms
Now, we can separate the terms in the series:
\[ \left( \frac{1}{1 \cdot 3} - \frac{2}{2 \cdot 4} + \frac{5}{5 \cdot 7} - \frac{6}{6 \cdot 8} + \ldots \right) + \left( \frac{3}{1 \cdot 3} - \frac{4}{2 \cdot 4} + \frac{7}{5 \cdot 7} - \frac{8}{6 \cdot 8} + \ldots \right) \]

### Step 3: Identify the Pattern
The first part of the series can be simplified:
\[ \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \ldots \]

This resembles the Taylor series expansion for \(\ln(1+x)\) evaluated at \(x=1\):
\[ \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \]

### Step 4: Evaluate the Series
By substituting \(x=1\):
\[ \ln(2) = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \ldots \]

Thus, the series converges to \(\ln(2)\).

### Final Answer
Therefore, the value of the given expression is:
\[ \ln(2) \]