If $|x| < 1$,then the coefficient of $x^n$ in the expansion of $(1 + x + x^2 + ....)^2$ will be

If $|x|$ is so small that $x^2$ and higher powers of $x$ may be neglected, then the approximate value of $\frac{\sqrt{4+x}+\sqrt[3]{8-x}}{\left(1-\frac{2x}{3}\right)^{\frac{3}{2}}}$ when $x=\frac{6}{25}$ is

$1 - \frac{1}{8} + \frac{1}{8} \cdot \frac{3}{16} - \frac{1 \cdot 3 \cdot 5}{8 \cdot 16 \cdot 24} + \dots =$

If the coefficient of $x^{13}$ in the expansion of $\frac{(1+x)^2}{(1-2x)^3}$ is $A \times 2^{10}$,then $A=$

The sum of the coefficients of $x^{-3/2}$ and $x^3$ in the expansion of $\sqrt{3+x} + \sqrt{5+x}$ when $3 < x < 5$ is

The sum of the series $1 + \frac{1}{5} + \frac{1 \times 3}{5 \times 10} + \frac{1 \times 3 \times 5}{5 \times 10 \times 15} + \dots$ is equal to