Answer
The Maclaurin polynomials:
$n=0$, ${\ \ \ }$ ${p_0}\left( x \right) = 0$
$n=1$, ${\ \ \ }$ ${p_1}\left( x \right) = x$
$n=2$, ${\ \ \ }$ ${p_2}\left( x \right) = x + {x^2}$
$n=3$, ${\ \ \ }$ ${p_3}\left( x \right) = x + {x^2} + \dfrac{1}{{2!}}{x^3}$
$n=4$, ${\ \ \ }$ ${p_4}\left( x \right) = x + {x^2} + \dfrac{1}{{2!}}{x^3} + \dfrac{1}{{3!}}{x^4}$
The $n$th Maclaurin polynomials in sigma notation:
$\mathop \sum \limits_{k = 1}^n \dfrac{{{x^k}}}{{\left( {k - 1} \right)!}}$
Work Step by Step
Let $f\left( x \right) = x{{\rm{e}}^x}$. Thus, the derivatives:
$f'\left( x \right) = x{{\rm{e}}^x} + {{\rm{e}}^x}$, ${\ \ \ \ \ }$ $f'\left( 0 \right) = 1$
$f{\rm{''}}\left( x \right) = x{{\rm{e}}^x} + 2{{\rm{e}}^x}$, ${\ \ \ \ \ }$ $f{\rm{''}}\left( 0 \right) = 2$
$f{\rm{'''}}\left( x \right) = x{{\rm{e}}^x} + 3{{\rm{e}}^x}$, ${\ \ \ \ \ }$ $f{\rm{'''}}\left( 0 \right) = 3$
$f{\rm{''''}}\left( x \right) = x{{\rm{e}}^x} + 4{{\rm{e}}^x}$, ${\ \ \ \ \ }$ $f{\rm{''''}}\left( 0 \right) = 4$
The Maclaurin polynomials:
$n=0$, ${\ \ \ }$ ${p_0}\left( x \right) = f\left( 0 \right) = 0$
$n=1$, ${\ \ \ }$ ${p_1}\left( x \right) = f\left( 0 \right) + f'\left( 0 \right)x = x$
$n=2$, ${\ \ \ }$ ${p_2}\left( x \right) = f\left( 0 \right) + f'\left( 0 \right)x + \dfrac{{f{\rm{''}}\left( 0 \right)}}{{2!}}{x^2} = x + {x^2}$
$n=3$, ${\ \ \ }$ ${p_3}\left( x \right) = f\left( 0 \right) + f'\left( 0 \right)x + \dfrac{{f{\rm{''}}\left( 0 \right)}}{{2!}}{x^2} + \dfrac{{f{\rm{'''}}\left( 0 \right)}}{{3!}}{x^3} = x + {x^2} + \dfrac{1}{{2!}}{x^3}$
$n=4$, ${\ \ \ }$ ${p_4}\left( x \right) = f\left( 0 \right) + f'\left( 0 \right)x + \dfrac{{f{\rm{''}}\left( 0 \right)}}{{2!}}{x^2} + \dfrac{{f{\rm{'''}}\left( 0 \right)}}{{3!}}{x^3} + \dfrac{{f{\rm{''''}}\left( 0 \right)}}{{4!}}{x^4} = x + {x^2} + \dfrac{1}{{2!}}{x^3} + \dfrac{1}{{3!}}{x^4}$
${p_{k + 1}}\left( x \right) = x + {x^2} + \dfrac{1}{{2!}}{x^3} + \dfrac{1}{{3!}}{x^4} + \cdot\cdot\cdot + \dfrac{{{x^{k + 1}}}}{{k!}}$, ${\ \ \ }$ for $k = 0,1,2,\cdot\cdot\cdot$
The $n$th Maclaurin polynomials for $x{{\rm{e}}^x}$ in sigma notation:
$\mathop \sum \limits_{k = 1}^n \dfrac{{{x^k}}}{{\left( {k - 1} \right)!}}$
