Tikz-PGF: Draw integral test plot

Question

So I'm doing a report on convergence tests in LaTeX, and would like to produce a diagram demonstrating the integral test for convergence. I am somewhat new to LaTeX, and a total n00b when it comes to Tikz, so any help would be much appreciated.

I am essentially trying to produce something like this:

You might ask why not just use this image, right? Truthfully, I would like to learn TikZ, and would very much appreciate any help in coming up with this plot.

Furthermore, I would like to place inside each bar the corresponding term of the series: a_1, a_2, a_3, ..., a_n

Any assistance would be much appreciated!

Answer 1

With tkz-fct and gnuplot

\documentclass[]{scrartcl}
\usepackage{tkz-fct}   

\begin{document}
\begin{tikzpicture}[scale=1.25]
\tkzInit[xmax=8,ymax=4]
\tkzAxeXY[ticks=false]
\tkzGrid   
\tkzFct[color = red, domain =0.125:8]{4./x}
\tkzDrawRiemannSumInf[fill=green!60,
                     opacity=.2,
                     color=green,
                     line width=1pt,
                     interval=1:8,
                     number=7] 
 \foreach \x/\t in {1.5/$a_1$,2.5/$a_2$,3.5/$a_3$,7.5/$a_7$}
 \node[green!50!black] at (\x,{4/(\x+1)-0.25}){\t};  
\end{tikzpicture}   
\end{document} 

Answer 2

The code from Jake's answer to Timing and integration with TikZ PGFlots discusses the mid-point rule. It can be tweaked to use right hand sums, and left hand sums.

% right hand sums
\pgfplotsset{
    right segments/.code={\pgfmathsetmacro\rightsegments{#1}},
    right segments=3,
    right/.style args={#1:#2}{
        ybar interval,
        domain=#1+((#2-#1)/\rightsegments):#2+((#2-#1)/\rightsegments),
        samples=\rightsegments+1,
        x filter/.code=\pgfmathparse{\pgfmathresult-((#2-#1)/\rightsegments)}
    }
}

% left hand sums
\pgfplotsset{
    left segments/.code={\pgfmathsetmacro\leftsegments{#1}},
    left segments=3,
    left/.style args={#1:#2}{
        ybar interval,
        domain=#1:#2,
        samples=\leftsegments+1,
        x filter/.code=\pgfmathparse{\pgfmathresult}
       }
}

and, finally, a complete MWE

\documentclass{article}

\usepackage{pgfplots}

% mid-point rule
\pgfplotsset{
    midpoint segments/.code={\pgfmathsetmacro\midpointsegments{#1}},
    midpoint segments=3,
    midpoint/.style args={#1:#2}{
        ybar interval,
        domain=#1+((#2-#1)/\midpointsegments)/2:#2+((#2-#1)/\midpointsegments)/2,
        samples=\midpointsegments+1,
        x filter/.code=\pgfmathparse{\pgfmathresult-((#2-#1)/\midpointsegments)/2}
    }
}

% right hand sums
\pgfplotsset{
    right segments/.code={\pgfmathsetmacro\rightsegments{#1}},
    right segments=3,
    right/.style args={#1:#2}{
        ybar interval,
        domain=#1+((#2-#1)/\rightsegments):#2+((#2-#1)/\rightsegments),
        samples=\rightsegments+1,
        x filter/.code=\pgfmathparse{\pgfmathresult-((#2-#1)/\rightsegments)}
    }
}

% left hand sums
\pgfplotsset{
    left segments/.code={\pgfmathsetmacro\leftsegments{#1}},
    left segments=3,
    left/.style args={#1:#2}{
        ybar interval,
        domain=#1:#2,
        samples=\leftsegments+1,
        x filter/.code=\pgfmathparse{\pgfmathresult}
       }
}



\begin{document}

% left hand sums
\begin{tikzpicture}[/pgf/declare function={f=4/x;}]
\begin{axis}[
        xmin=0,xmax=9,ymin=0,ymax=4,
    domain=0:10,
    samples=100,
    axis lines=middle
]
\addplot [thick, red] {f};
\addplot [
    black!80,fill=green,opacity=.3,
    left segments=7,
    left=1:8
] {f};
\end{axis}
\end{tikzpicture}

% right hand sums
\begin{tikzpicture}[/pgf/declare function={f=4/x;}]
\begin{axis}[
        xmin=0,xmax=9,ymin=0,ymax=4,
    domain=0:10,
    samples=100,
    axis lines=middle
]
\addplot [thick, red] {f};
\addplot [
    black!80,fill=green,opacity=.3,
    right segments=7,
    right=1:8,
] {f};
\end{axis}
\end{tikzpicture}

% mid point
\begin{tikzpicture}[/pgf/declare function={f=4/x;}]
\begin{axis}[
        xmin=0,xmax=8,ymin=0,ymax=4,
    domain=0:10,
    samples=100,
    axis lines=middle
]
\addplot [thick,red] {f};
\addplot [
    black!80,fill=green,opacity=.3,
    midpoint segments=7,
    midpoint=1:8,
] {f};
\end{axis}
\end{tikzpicture}


\end{document}