The coefficient of `x^(n)` in the expansion of `log_(e ) ((1)/(1+x+x^(2)+x^(3)))`, when n is odd is

Options

A. `-(2)/(n)`
B. `-(1)/(n)`
C. `(1)/(n)`
D. None of these

Answer

To find the coefficient of \( x^n \) in the expansion of \( \log_e \left( \frac{1}{1 + x + x^2 + x^3} \right) \) when \( n \) is odd, we can follow these steps:

### Step 1: Rewrite the logarithm
We start with the expression:
\[ \log_e \left( \frac{1}{1 + x + x^2 + x^3} \right) = -\log_e(1 + x + x^2 + x^3) \]

### Step 2: Factor the denominator
Next, we can factor the polynomial \( 1 + x + x^2 + x^3 \):
\[ 1 + x + x^2 + x^3 = (1 + x)(1 + x^2) \]
Thus, we have:
\[ -\log_e(1 + x + x^2 + x^3) = -\log_e((1 + x)(1 + x^2)) \]

### Step 3: Apply logarithmic properties
Using the property of logarithms, we can separate the terms:
\[ -\log_e((1 + x)(1 + x^2)) = -\log_e(1 + x) - \log_e(1 + x^2) \]

### Step 4: Expand using Taylor series
Now, we can expand both logarithmic terms using the Taylor series expansion:
\[ \log_e(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]
\[ \log_e(1 + x^2) = x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \frac{x^8}{4} + \cdots \]

### Step 5: Combine the expansions
Thus, we have:
\[ -\log_e(1 + x) = -\left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots\right) \]
\[ -\log_e(1 + x^2) = -\left(x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \frac{x^8}{4} + \cdots\right) \]

Combining these, we get:
\[ -\log_e(1 + x + x^2 + x^3) = -\left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots\right) - \left(x^2 - \frac{x^4}{2} + \frac{x^6}{3} - \cdots\right) \]

### Step 6: Identify the coefficient of \( x^n \)
Now, we need to find the coefficient of \( x^n \) when \( n \) is odd. The odd powers of \( x \) from the first expansion contribute:
- From \( -x \): coefficient is \(-1\)
- From \( \frac{x^3}{3} \): coefficient is \(\frac{1}{3}\)
- From higher odd powers, we can see that the coefficients will follow the pattern of \(-\frac{1}{n}\).

Thus, for odd \( n \):
\[ \text{Coefficient of } x^n = -\frac{1}{n} \]

### Final Answer
The coefficient of \( x^n \) in the expansion of \( \log_e \left( \frac{1}{1 + x + x^2 + x^3} \right) \) when \( n \) is odd is:
\[ -\frac{1}{n} \]