# Customize options of Taylor polynomials

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=$y$]
\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)}
}
\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?

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 }]
\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)}
\addlegendentryexpanded{order $\the\numexpr2*\nn+1$} • @BambOo This or maybe the sum function from here. – user194703 May 4 at 16:38
• @BambOo I think that the idea of accumulating macros/stuff in this way is rather old. There are very nice answers by Jake who used some pgfplotstable functions to build up such sums. (And I had upvoted your question some time ago, but at that point I did not understand how powerful \pgfmathdeclarefunction really is. These functions can be used everywhere where TikZ parses expressions, also in coordinates and so on.) – user194703 May 4 at 16:44