# How to use a summation in a tikz plot

I need to plot this function using tikz, but I have no idea how to go about using the summation in tikz. If someone could help me I would really appreciate it. The code I have is this:

\documentclass{article}

\usepackage{pgfplots}
\usepackage{tikz}

\usepackage[paperwidth=20cm, paperheight=10cm, margin=0.0cm]{geometry}

\begin{document}
\hspace{-0.4cm}
\begin{tikzpicture}[
declare function={p(\k,\n) =((\n!*(\k*(\k-1)))/((\n^(\k))*((\n-(\k-
1))!)));}
]
\begin{axis}[axis lines=middle, grid=both, width=20cm, height=10cm, grid
style={line width=.2pt, draw=gray!10},major grid style={line
width=.3pt,draw=gray!50},
xlabel=$k$, ylabel=$y$,
ylabel style={anchor=south},
xlabel style={anchor=west},
ymax=1, xmax=21.5,
domain=0:21.5, samples at={0,...,21},
]
\addplot+ [blue, thick, mark size={1.6pt}, mark options={draw=blue,
fill=white!75!cyan}] {p(x,20)};
\end{axis}
\end{tikzpicture}

\end{document}


This plots the function of x, but I don't know how to sum the function from k=2 to x+1.

• possibly helpful? tex.stackexchange.com/questions/291404/… – cmhughes Nov 23 '17 at 15:01
• Thanks, I saw that earlier but couldn't figure it out. I've only been using latex for a short time and Im struggling to work out what to do. – John 545 Nov 23 '17 at 15:08
• Is this not possible in tikz? – John 545 Nov 23 '17 at 20:03

You can use Lua to compute the sum. Then you have to typeset using LuaLaTeX of course.

\documentclass{article}

\usepackage{pgfplots}

\usepackage{luacode}
\begin{luacode*}
function factorial(n)
assert(n >= 0, "Factorial is only valid for positive integers")
if n == 0 then
return 1
end
return n*factorial(n-1)
end

function p(x)
assert(x == math.floor(x), "x must be an integer")
res = 0
for k = 2, x+1 do
res = res + factorial(x)/factorial(x-(k-1)) * k*(k-1)/(x^k)
end
tex.sprint(res)
end
\end{luacode*}

\begin{document}

\begin{tikzpicture}[
declare function={p(\n) = \directlua{p(\n)};}
]
\begin{axis}[
use fpu=false, % very important!
xlabel=$x$, ylabel=$p(x)$,
samples at={0,...,21},
only marks,
]