Question Number 9804 by FilupSmith last updated on 05/Jan/17
$$\mathrm{Express}\:{f}\left({x}\right)={x}^{{t}} {e}^{{x}} \:\mathrm{as}\:\mathrm{a}\:\mathrm{series} \\ $$
Commented by FilupSmith last updated on 07/Jan/17
$${x}^{{t}} ={e}^{{t}\mathrm{ln}\left({x}\right)} \\ $$$$\:\:\:\:=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{t}^{{n}} \mathrm{ln}^{{n}} \left({x}\right)}{{n}!} \\ $$$$\therefore\:{x}^{{t}} {e}^{{x}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{t}^{{n}} {e}^{{x}} \mathrm{ln}^{{n}} \left({x}\right)}{{n}!} \\ $$