If $x = \frac{1}{5} + \frac{1 \times 3}{5 \times 10} + \frac{1 \times 3 \times 5}{5 \times 10 \times 15} + \ldots$,then $3x^2 + 6x =$

The coefficient of $x^3$ in the power series expansion of $\frac{1+4x-3x^2}{(1+3x)^3}$ is

The expression $\frac{1}{(x^2 + \frac{1}{x})^{4/3}}$ can be expanded by the binomial theorem if:

If $-\frac{2}{3} < x < \frac{2}{3}$,then the value of the $5^{\text{th}}$ term in the expansion of $\frac{1}{\sqrt[3]{2-3x}}$ when $x=\frac{1}{2}$ is

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 $\alpha = \frac{5}{2! \times 3} + \frac{5 \times 7}{3! \times 3^2} + \frac{5 \times 7 \times 9}{4! \times 3^3} + \ldots$,then $\alpha^2 + 4\alpha =$