If $n$ is a positive integer and the coefficient of $x^{10}$ in the expansion of $(1+x)^{15}$ is equal to the coefficient of $x^5$ in the expansion of $(1-x)^{-n}$,then $n=$

The sum of the series $1+\frac{2}{3}\left(\frac{1}{8}\right)+\frac{2 \times 5}{3 \times 6}\left(\frac{1}{8}\right)^2+\frac{2 \times 5 \times 8}{3 \times 6 \times 9}\left(\frac{1}{8}\right)^3+\ldots$ is

If $x$ is small,so that $x^2$ and higher powers can be neglected,then the approximate value for $\frac{(1-2 x)^{-1}(1-3 x)^{-2}}{(1-4 x)^{-3}}$ is

The sum of the series $\frac{3}{4 \cdot 8}-\frac{3 \cdot 5}{4 \cdot 8 \cdot 12}+\frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12 \cdot 16}-\ldots$

If $x > \sqrt{3}$ and $\frac{x^2+1}{(x^2+2)(x^2+3)}$ is expanded in terms of powers of $x^{-2}$,then the coefficient of $x^{-8}$ is

The coefficient of $x^3$ in the expansion of $(1-x)^{3/2}$,$(|x| < 1)$ is