Question

Find the coefficient of $x^2$ in the expansion of $e^{2x+3}$ as a series in powers of $x.$

Answer 1

In the exponential series $e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^2}{3!}+...$
Hence $e^{2x+3}=e^3.e^{2x}=e^3[1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^2}{3!}+...]$
Thus, the coefficient of $x^2$ in the expansion of $e^{2x+3}$ is $e^3.\frac{2^2}{2!}=2e^3.$