If $y = x - \frac{x^2}{2!} + \frac{x^3}{3!} - \frac{x^4}{4!} + \dots$,then $x = $

  • A
    $\log_e(1 - y)$
  • B
    $\frac{1}{\log_e(1 - y)}$
  • C
    $\log_e\left(\frac{1}{1 - y}\right)$
  • D
    $\log_e(1 + y)$

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The coefficient of $x^n$ in the expansion of $\log_e(1 + 3x + 2x^2)$ is

$\frac{m - n}{m + n} + \frac{1}{3}\left( \frac{m - n}{m + n} \right)^3 + \frac{1}{5}\left( \frac{m - n}{m + n} \right)^5 + \dots \infty = $

If $\alpha, \beta$ are the roots of the equation $x^2 - px + q = 0$,then $\log_e(1 + px + qx^2) = $

If $0 < y < 2^{1/3}$ and $x(y^3 - 1) = 1$,then $\frac{2}{x} + \frac{2}{3x^3} + \frac{2}{5x^5} + \dots$ is equal to:

$\frac{1}{2}x^2 + \frac{2}{3}x^3 + \frac{3}{4}x^4 + \dots \infty = $

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