If `|x|lt 1`, tnen the coefficient of `x^(5)` in the expansion of `(1-x)log_(e )(1-x)` is

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(20)`
D. `(1)/(10)`

Answer

To find the coefficient of \( x^5 \) in the expansion of \( (1 - x) \log_e(1 - x) \), we can follow these steps:

### Step 1: Expand \( \log_e(1 - x) \)

The Taylor series expansion for \( \log_e(1 - x) \) around \( x = 0 \) is given by:

\[ \log_e(1 - x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \frac{x^5}{5} - \ldots \]

### Step 2: Multiply by \( (1 - x) \)

Now, we need to multiply \( (1 - x) \) by the series we just obtained:

\[ (1 - x) \log_e(1 - x) = (1 - x) \left( -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \frac{x^5}{5} - \ldots \right) \]

### Step 3: Distribute \( (1 - x) \)

Distributing \( (1 - x) \) gives us:

\[ = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \frac{x^5}{5} + x^2 + \frac{x^3}{2} + \frac{x^4}{3} + \frac{x^5}{4} + \ldots \]

### Step 4: Combine like terms

Now, we need to combine the terms that contribute to \( x^5 \):

1. From \( -\frac{x^5}{5} \)
2. From \( \frac{x^5}{4} \)

Thus, the coefficient of \( x^5 \) is:

\[ -\frac{1}{5} + \frac{1}{4} \]

### Step 5: Find a common denominator

To combine these fractions, we find a common denominator, which is 20:

\[ -\frac{1}{5} = -\frac{4}{20}, \quad \frac{1}{4} = \frac{5}{20} \]

### Step 6: Combine the fractions

Now, adding these together:

\[ -\frac{4}{20} + \frac{5}{20} = \frac{1}{20} \]

### Conclusion

Thus, the coefficient of \( x^5 \) in the expansion of \( (1 - x) \log_e(1 - x) \) is:

\[ \frac{1}{20} \]

### Final Answer

The answer is \( \frac{1}{20} \).

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