If x<1, then the coefficient of x5 in the expansion of (1 – x) loge(1 – x) is

(1-x) loge(1-x)=(1-x) -x-x22-x33-x44-x55-....x5 Coef=1-15--14 =-4+520=120

If x<1, then the coefficient of x5 in the expansion of (1 – x) loge(1 – x) is

(1-x) loge(1-x)=(1-x) -x-x22-x33-x44-x55-....x5 Coef=1-15--14 =-4+520=120

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If $|x| < 1,$ then the coefficient of x⁵ in the expansion of (1 – x) log_e(1 – x) is

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$\frac{1}{20}$

$\frac{1}{2}$

$\frac{1}{10}$

$\frac{1}{4}$

answer is C.

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Detailed Solution

$\begin{array}{cl} (1 - x) \log_{e}^{(1 - x)} \\ = & (1 - x) (- x - \frac{x^{2}}{2} - \frac{x^{3}}{3} - \frac{x^{4}}{4} - \frac{x^{5}}{5} - . . . .) \end{array}$
$\begin{array}{rcl} x^{5} Coef & = & 1 (\frac{- 1}{5}) - (\frac{- 1}{4}) \\ = & \frac{- 4 + 5}{20} = \frac{1}{20} \end{array}$