If $P_n$ denotes the product of the binomial coefficients in the expansion of $(1+x)^n$, then $\frac{P_{n+1}}{P_n}=$

The coefficient of $x^3$ in the expansion of $\frac{x^4+1}{\left(x^2+1\right)(x-1)}$ when it is expressed in terms of positive integral powers of $x$, is

If $(1+x)^n=\sum_{r=0}^n C, x^r$, then the value of $C_0+\left(C_0+C_1\right)+\left(C_0+C_1+C_2\right)+\ldots+ \left(C_0+C_1+C_2+\ldots+C_n\right)$ is

If $x$ is so large that terms containing $x^{-3}, x^{-4}, x^{-5}, \ldots$ can be neglected, then the approximate value of $\left(\frac{3 x-5}{4 x^2+3}\right)^{-1 / 5}$ is