`-2[(1)/(8)+(1)/(64)+(1)/(384)+…..oo]=`

Options

A. `log_(e )(1//4)`
B. `log_(e )(1//2)`
C. `log_(e )(3//4)`
D. `log_(e )(3//8)`

Answer

To solve the given series \(-2\left(\frac{1}{8} + \frac{1}{64} + \frac{1}{384} + \ldots\right)\), we can follow these steps:

### Step 1: Identify the series
The series inside the brackets is:
\[ S = \frac{1}{8} + \frac{1}{64} + \frac{1}{384} + \ldots \]

### Step 2: Rewrite the terms
We can express the terms in a more recognizable form. Notice that:
- \(\frac{1}{8} = \frac{1}{2^3}\)
- \(\frac{1}{64} = \frac{1}{2^6}\)
- \(\frac{1}{384} = \frac{1}{2^8 \cdot 3}\)

The denominators suggest a pattern involving powers of 2 and possibly factorials or products.

### Step 3: Factor out common terms
Let's factor out the common terms from the series. We can rewrite the series as:
\[ S = \frac{1}{8} \left(1 + \frac{1}{8} + \frac{1}{48} + \ldots\right) \]

### Step 4: Recognize the series as a logarithmic series
The series can be expressed in terms of a logarithmic expansion. Recall the expansion:
\[ \log(1-x) = -\left(x + \frac{x^2}{2} + \frac{x^3}{3} + \ldots\right) \]
If we set \(x = \frac{1}{4}\), we can relate our series to this logarithmic form.

### Step 5: Apply the logarithmic series
Thus, we can express \(S\) as:
\[ S = -\log\left(1 - \frac{1}{4}\right) = -\log\left(\frac{3}{4}\right) \]

### Step 6: Multiply by -2
Now, we multiply the result by -2:
\[ -2S = -2\left(-\log\left(\frac{3}{4}\right)\right) = 2\log\left(\frac{3}{4}\right) \]

### Final Answer
Thus, the final answer is:
\[ -2\left(\frac{1}{8} + \frac{1}{64} + \frac{1}{384} + \ldots\right) = 2\log\left(\frac{3}{4}\right) \]