Find the coefficient of $x^{24}$ in the power series of $e^{2x}(e^x-1)$.

Question

I tried to work through the problem algebraically, and got to $\frac{2^{24}}{24!}(2^{24}-1)$, but comparing with the Taylor series generated for the function by Wolfram Alpha, it seems to be incorrect. I plugged in the power series for $e^x$ and used the formula for multiplication of series.

As a follow up question, is a power series converging to a function singular? Or can there be multiple correct answers?

Answer 1

Just write $e^{2x}(e^{x}-1)$ as $e^{3x}-e^{2x}$. The coefficident is $\frac {3^{24}} {(24)!} -\frac {2^{24}} {(24)!}$.