In the expansion of $2 \ln x - \ln(x + 1) - \ln(x - 1)$,the coefficient of $x^{-4}$ is

Evaluate: $\log_e \sqrt{\frac{1+x}{1-x}}$

$\frac{4}{1 \times 3} - \frac{6}{2 \times 4} + \frac{12}{5 \times 7} - \frac{14}{6 \times 8} + \dots \infty = $

If $0 < a, b < 1$ and $\tan^{-1} a + \tan^{-1} b = \frac{\pi}{4}$,then the value of $(a+b) - \left(\frac{a^2+b^2}{2}\right) + \left(\frac{a^3+b^3}{3}\right) - \left(\frac{a^4+b^4}{4}\right) + \dots$ is ..... .

If $x, y, z$ are three consecutive positive integers,then $\frac{1}{2}\log_e x + \frac{1}{2}\log_e z + \frac{1}{2xz + 1} + \frac{1}{3}\left( \frac{1}{2xz + 1} \right)^3 + \dots = $

If $P = 1 + \frac{1}{2 \times 2} + \frac{1}{3 \times 2^{2}} + \dots$ and $Q = \frac{1}{1 \times 2} + \frac{1}{3 \times 4} + \frac{1}{5 \times 6} + \dots$,then