Assertion (A) : The coefficient of `x^(5)` in the expansion `log_(e ) ((1+x)/(1-x))` is `(2)/(5)`
Reason (R ) : The equality
`log((1+x)/(1-x))=2[x+(x^(2))/(2)+(x^(3))/(3)+(x^(4))/(4)+….oo]` is valid for `|x| lt 1`

Options

A. A is true, R is true and R is correct explanation of A
B. A is true, R is true and R is not correct explanation of A
C. A is true, R is false
D. A is false, R is true

Answer

To solve the problem, we need to analyze both the assertion (A) and the reason (R) given in the question.

### Step 1: Understanding the Assertion
The assertion states that the coefficient of \( x^5 \) in the expansion of \( \log_e \left( \frac{1+x}{1-x} \right) \) is \( \frac{2}{5} \).

### Step 2: Expanding the Logarithm
We start with the expression:
\[ \log_e \left( \frac{1+x}{1-x} \right) \]
Using the property of logarithms, we can rewrite this as:
\[ \log_e(1+x) - \log_e(1-x) \]

### Step 3: Using Taylor Series Expansion
The Taylor series expansion for \( \log_e(1+x) \) around \( x=0 \) is:
\[ \log_e(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]
And for \( \log_e(1-x) \):
\[ \log_e(1-x) = -\left( x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \cdots \right) \]

### Step 4: Combining the Series
Now, substituting these expansions back into our expression:
\[ \log_e(1+x) - \log_e(1-x) = \left( x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \right) - \left( -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \cdots \right) \]
This simplifies to:
\[ \log_e \left( \frac{1+x}{1-x} \right) = 2x + \frac{2x^3}{3} + \frac{2x^5}{5} + \cdots \]

### Step 5: Identifying the Coefficient of \( x^5 \)
From the expansion, we can see that the coefficient of \( x^5 \) is \( \frac{2}{5} \). Thus, the assertion (A) is **true**.

### Step 6: Analyzing the Reason
The reason (R) states that:
\[ \log \left( \frac{1+x}{1-x} \right) = 2 \left[ x + \frac{x^2}{2} + \frac{x^3}{3} + \frac{x^4}{4} + \cdots \right] \]
However, we have derived that:
\[ \log \left( \frac{1+x}{1-x} \right) = 2 \left[ x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots \right] \]
The series does not include \( \frac{x^2}{2} \) and \( \frac{x^4}{4} \), which makes the reason (R) **false**.

### Conclusion
Thus, the assertion is true, and the reason is false. The correct option is:
- A is true, R is false.