Let $S_{k}$ be the sum of an infinite $GP$ series whose first term is $k$ and common ratio is $\frac{k}{k+1}$ $(k>0)$. Then, the value of $\sum_{k=1}^{\infty} \frac{(-1)^{k}}{S_{k}}$ is equal to

The sum of the series $\log_{4} 2 - \log_{8} 2 + \log_{16} 2 - \dots$ is

If $y = 2x^2 - 1$,then $\left[ \frac{1}{y} + \frac{1}{3y^3} + \frac{1}{5y^5} + \dots \right]$ is equal to

If $1+\frac{\sqrt{3}-\sqrt{2}}{2 \sqrt{3}}+\frac{5-2 \sqrt{6}}{18}+\frac{9 \sqrt{3}-11 \sqrt{2}}{36 \sqrt{3}}+\frac{49-20 \sqrt{6}}{180}+\ldots$ upto $\infty = 2\left(\sqrt{\frac{b}{a}}+1\right) \log _e\left(\frac{a}{b}\right)$,where $a$ and $b$ are integers with $\operatorname{gcd}(a, b)=1$,then $11 a+18 b$ is equal to ...............

$\frac{1}{1 \cdot 3} + \frac{1}{2} \cdot \frac{1}{3 \cdot 5} + \frac{1}{3} \cdot \frac{1}{5 \cdot 7} + \dots \infty = $

The expression $\log_{e} 2 + \log_{e} \left( 1 + \frac{1}{2} \right) + \log_{e} \left( 1 + \frac{1}{3} \right) + \dots + \log_{e} \left( 1 + \frac{1}{n - 1} \right)$ is equal to