Question Number 30754 by abdo imad last updated on 25/Feb/18
$${prove}\:{that}\:{e}^{{x}} =\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{{x}^{{k}} }{{k}!}\:+\frac{{x}^{{n}+\mathrm{1}} }{{n}!}\:\int_{\mathrm{0}} ^{} \left(\mathrm{1}−{t}\right)^{{n}} \:{e}^{{tx}} {dt} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\:{e}^{{x}} =\:\sum_{{k}=\mathrm{0}} ^{\infty\:} \:\:\frac{{x}^{{k}} }{{k}!}\:. \\ $$