Customize options of Taylor polynomials

Question

I am doing the Taylor series of sin function by using this code:

\documentclass{article}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}
 \begin{axis}[domain=-3.14:3.14,samples=100,smooth,no markers,axis lines=middle,ymax=2,ymin=-2,xlabel=$x$,ylabel=$y$]
  \addplot[thick,color=orange,domain=-3.14:3.14] {sin(deg(x))};
  \def\myfun{0}
  \pgfplotsforeachungrouped \nn in {0,1,2,3,4}
  {\edef\myfun{\myfun+((-1)^(\nn))*pow(x,2*\nn+1)/factorial(2*\nn+1)}
   \addplot+{\myfun};
  }
 \end{axis} 
\end{tikzpicture}
\end{document}

I would like to customise the options of colors in each new term inserted in the series (tones of blue), specify the width of line, etc.... Moreover, I would like to insert a legend each time a new parcell is inserted. How can I do this, please?

Answer 1

As explained in the section Defining Cycle Lists based on Color Maps that starts on p. 220 of the pgfplots manual v1.17, you can create a cycle list based on a colormap. I added something that interpolates between blue and cyan, but you can change this, of course. And you can add the legend entries with \addlegendentryexpanded.

\documentclass[tikz,border=3mm]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.16}
\begin{document}
\begin{tikzpicture}
 \begin{axis}[width=10cm,domain=-pi:pi,samples=101,smooth,
 no markers,axis lines=middle,
 legend style={at={(0.75,0.4)},anchor=north},
 ymax=2,ymin=-2,xlabel=$x$,ylabel=$y$,
 colormap={blueblack}{color=(blue) color=(cyan)}, 
 cycle multiindex* list={[samples of colormap=6]\nextlist
    mark list\nextlist }]
  \addplot[thick,color=orange,domain=-pi:pi] {sin(deg(x))};
  \addlegendentry{$\sin x$}
  \edef\myfun{0}
  \pgfplotsforeachungrouped \nn in {0,1,2,3,4}
  {\edef\myfun{\myfun+((-1)^(\nn))*pow(x,2*\nn+1)/factorial(2*\nn+1)}
   \addplot+{\myfun};
   \addlegendentryexpanded{order $\the\numexpr2*\nn+1$}
  }
 \end{axis} 
\end{tikzpicture}
\end{document}