If `"log"_(e) (1)/(1 + x + x^(2) + x^(3))` be expemnded in asecending power of x , show that the coefficient of ` x^(n)`
is `- (1)/(n)` if n is odd or of the form 4m + 2 and `(3)/(n)`if n is of the form 4m

Explanation

To solve the problem, we need to expand the expression \( \log_e \left( \frac{1}{1 + x + x^2 + x^3} \right) \) in ascending powers of \( x \) and find the coefficients of \( x^n \) for different values of \( n \).

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

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