\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{0mm}

\usepackage{pgfplots}
\usepackage{tikz}
\usetikzlibrary{arrows, positioning, calc}

\usepackage{xcolor}
\definecolor{pink}{HTML}{FF0BB7}

\begin{document}
\begin{preview}
\begin{tikzpicture}
    \begin{axis}[
        axis x line=middle,
        axis y line=middle,
        enlarge y limits=true,
        xmin=-8.5, xmax=8.5, % Positive domain...
        width=15cm, height=8cm,     % size of the image
        grid = major,
        grid style={dashed, gray!30},
        ymin=-4,      % start the diagram at this y-coordinate
        ymax= 4,      % end   the diagram at this y-coordinate
        axis background/.style={fill=white},
        ylabel=$y$,
        xlabel=$x$,
        legend style={at={(0.07,0.37)}, anchor=north}
     ]
      \addplot[domain=-8:8,thick,samples=200,red]   {x};
      \addplot[domain=-8:8,thick,samples=200,green] {x-x^3/6};
      \addplot[domain=-8:8,thick,samples=200,blue]  {x-x^3/6+x^5/120};
      \addplot[domain=-8:8,thick,samples=200,pink]  {x-x^3/6+x^5/120-x^7/(7!)+x^9/(9!)-x^11/(11!)+x^13/(13!)-x^15/(15!)};
      \addplot[domain=-8:8,thick,samples=200,black] {sin(deg(x))};
      \legend{$T_1$, $T_3$, $T_5$, $T_{15}$, $T_\infty$}
    \end{axis} 
\end{tikzpicture}
\end{preview}
\end{document}
