Sum inside addplot call

Here is with xinttools. I suspect (but did not check) unexpandable loop could not be used inside \addplot argument, so I defined preliminary macro \myterm to use with the expandable utilities \xintApply and \xintListWithSep.

The external \xintFor is not expandable, and dispense us of any preliminary macro definition.

\documentclass[tikz,border=10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.5}
\usepgfplotslibrary{colorbrewer}
\pgfplotsset{cycle list/Reds-6}
\usepackage{xinttools}
\begin{document}

\begin{tikzpicture}
\def\myterm#1{sin(4*#1.5*\x r)/#1.5}
\begin{axis}[%
    width=\textwidth,
    axis x line=bottom,
    axis y line=left,cycle multi list={Reds-6}]
    \xintFor #1 in {0, 1, 2, 3, 4, 5} \do 
    {%    
    \addplot+[samples=300,smooth] 
     {\xintListWithSep{+}{\xintApply\myterm{\xintSeq{0}{#1}}}}; 
    }   
\end{axis}
\end{tikzpicture}
\end{document}

I set up a stencil and used listofitems to parse it for ? via

\readlist\termstencil{sin(4*?.5*\x r)/?.5}

Then, substitute for ? each time through the loop, a value of 0, then 1, ..., 5. A few checks had to be made not to add a + before the first term, etc. The resulting tokens are collected into the macro \myeqn, which is then passed to \addplot.

\documentclass[tikz,border=10pt]{standalone}
\usepackage{pgfplots}
\pgfplotsset{compat=1.5}
\usepgfplotslibrary{colorbrewer}
\pgfplotsset{cycle list/Reds-6}
\usepackage{listofitems}
\makeatletter
\newcommand\loopthroughterm[1]{%
  \def\myeqn{}%
  \foreach\z in{0,...,#1}{%
    \ifnum\z=0\relax\else\g@addto@macro\myeqn{+}\fi%
    \foreachitem\zz\in\termstencil[]{%
      \ifnum\zzcnt=1\else%
        \expandafter\g@addto@macro\expandafter\myeqn\expandafter{\z}\fi%
      \expandafter\g@addto@macro\expandafter\myeqn\expandafter{\zz}%
    }%
  }%
}
\makeatother
\begin{document}
\begin{tikzpicture}
\begin{axis}[%
    width=\textwidth,
    axis x line=bottom,
    axis y line=left,cycle multi list={Reds-6}]
    \setsepchar{?}
    \readlist\termstencil{sin(4*?.5*\x r)/?.5}
    \foreach\k in{0,...,5}{
      \loopthroughterm{\k}
      \addplot+[samples=300,smooth] {\myeqn};
    }
\end{axis}
\end{tikzpicture}
\end{document}

Upon exiting the tikzpicture, one can \detokenize\expandafter{\myeqn} to confirm the tokens themselves used for the final loop index:

---

ADDENDUM

Seeing jbfu's use of a helper macro, I realized I could use the same concept to dispense with the listofitems approach, and not use another package in its place.

Basically, I get the \loopthroughterm macro to create a succession of tokens of the form \termstencil{0}+\termstencil{1}+\termstencil{2}.... Then, as long as \termstencil is defined by the user properly, in this case as

\def\termstencil#1{sin(4*#1.5*\x r)/#1.5}

then all works out in the end.

\documentclass[tikz,border=10pt]{standalone}
\usepackage{pgfplots}
\usepackage[T1]{fontenc}
\pgfplotsset{compat=1.5}
\usepgfplotslibrary{colorbrewer}
\pgfplotsset{cycle list/Reds-6}
\makeatletter
\newcommand\loopthroughterm[1]{%
  \def\myeqn{}%
  \foreach\z in{0,...,#1}{%
    \ifnum\z=0\relax\else\g@addto@macro\myeqn{+}\fi%
    \expandafter\g@addto@macro\expandafter\myeqn\expandafter{%
      \expandafter\termstencil\expandafter{\z}}%
  }%
}
\makeatother
\begin{document}
\begin{tikzpicture}
\begin{axis}[%
    width=\textwidth,
    axis x line=bottom,
    axis y line=left,cycle multi list={Reds-6}]
    \def\termstencil#1{sin(4*#1.5*\x r)/#1.5}
    \foreach\k in{0,...,5}{
      \loopthroughterm{\k}
      \addplot+[samples=300,smooth] {\myeqn};
    }
\end{axis}
\end{tikzpicture}
\end{document}

The plot is the same as in the prior code, but the detokenized form of \myeqn is now this:

Additionally, with the use of a few extra \expandafters, in the form of

\newcommand\loopthroughterm[1]{%
  \def\myeqn{}%
  \foreach\z in{0,...,#1}{%
    \ifnum\z=0\relax\else\g@addto@macro\myeqn{+}\fi%
    \expandafter\expandafter\expandafter\g@addto@macro\expandafter%
      \expandafter\expandafter\myeqn\expandafter\expandafter\expandafter{%
        \expandafter\termstencil\expandafter{\z}}%
  }%
}

the final \myeqn now contains the actual tokens desired: