The series expansion of `log{(1+x)^(1+x)(1-x)^(1-x)}` is

Options

A. `2{(x^(2))/(1.2)+(x^(4))/(3.4)+(x^(6))/(5.6)+..}`
B. `{(x^(2))/(1.2)+(x^(4))/(3.4)+(x^(6))/(5.6)+..}`
C. `2{(x^(2))/(1.2)+(x^(4))/(2.3)+(x^(6))/(3.4)+..}`
D. none of these

Answer

To find the series expansion of the expression \( \log{(1+x)^{1+x}(1-x)^{1-x}} \), we can follow these steps:

### Step 1: Apply Logarithmic Properties
Using the properties of logarithms, we can separate the expression:
\[ \log{(1+x)^{1+x}(1-x)^{1-x}} = \log{(1+x)^{1+x}} + \log{(1-x)^{1-x}} \]
Using the power property of logarithms, we can rewrite this as:
\[ (1+x)\log{(1+x)} + (1-x)\log{(1-x)} \]

### Step 2: Expand the Logarithmic Functions
Next, we need to use the Taylor series expansion for \( \log{(1+x)} \) and \( \log{(1-x)} \):
\[ \log{(1+x)} = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]
\[ \log{(1-x)} = -\left(x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \cdots\right) \]

### Step 3: Substitute the Expansions
Now, substitute these expansions back into our expression:
\[ (1+x)\left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots\right) + (1-x)\left(-x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \cdots\right) \]

### Step 4: Distribute and Combine Like Terms
Distributing the terms gives us:
\[ (1+x)\left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\right) + (1-x)\left(-x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4}\right) \]
This results in:
\[ x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + x^2 - \frac{x^3}{2} + \frac{x^4}{3} - x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} \]

### Step 5: Simplify the Expression
Now, we combine like terms:
- The \( x \) terms cancel out.
- The \( x^2 \) terms: \( -\frac{x^2}{2} + x^2 - \frac{x^2}{2} = 0 \)
- The \( x^3 \) terms: \( \frac{x^3}{3} - \frac{x^3}{2} - \frac{x^3}{3} = -\frac{x^3}{2} \)
- The \( x^4 \) terms: \( -\frac{x^4}{4} + \frac{x^4}{3} - \frac{x^4}{4} = -\frac{x^4}{6} \)

Thus, the series expansion simplifies to:
\[ -\frac{x^3}{2} - \frac{x^4}{6} + \cdots \]

### Final Result
The series expansion of \( \log{(1+x)^{1+x}(1-x)^{1-x}} \) is:
\[ -\frac{x^3}{2} - \frac{x^4}{6} + \cdots \]