\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Use 6 rectangles to approximate \(\displaystyle \int\limits_{1}^{5} \ln{\left(x \right)}\, dx\) using the left endpoint. Draw a graph showing the sum.
\(\displaystyle \Delta x = \frac{b-a}{n}=\frac{2}{3}\) and \(\displaystyle x_i = a+i\Delta x = \frac{2 i}{3} + 1\). Drawing a graph gives:
\begin{tikzpicture}[yscale=2]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-1,-1) grid (6,2.10000000000000);
\draw[latex-latex, very thick] (-1.4,0)--(6.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-1.1)--(0,2.2) node[above]{$y$};
\foreach \x in { -1,1,...,6 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -1,-0.5,...,2.10000000000000 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={1.0}:{5.0}, samples=200, samples =350] plot(\x, {ln((\x))});
\draw[fill=blue, opacity=0.3](1.00, 0) rectangle (1.67, 0);
\draw[thick](1.00, 0) rectangle (1.67, 0);
\draw[fill=blue, opacity=0.3](1.67, 0.511) rectangle (2.33, 0);
\draw[thick](1.67, 0.511) rectangle (2.33, 0);
\draw[fill=blue, opacity=0.3](2.33, 0.847) rectangle (3.00, 0);
\draw[thick](2.33, 0.847) rectangle (3.00, 0);
\draw[fill=blue, opacity=0.3](3.00, 1.10) rectangle (3.67, 0);
\draw[thick](3.00, 1.10) rectangle (3.67, 0);
\draw[fill=blue, opacity=0.3](3.67, 1.30) rectangle (4.33, 0);
\draw[thick](3.67, 1.30) rectangle (4.33, 0);
\draw[fill=blue, opacity=0.3](4.33, 1.47) rectangle (5.00, 0);
\draw[thick](4.33, 1.47) rectangle (5.00, 0);
\end{tikzpicture}
\begin{question}Use 6 rectangles to approximate $\int\limits_{1}^{5} \ln{\left(x \right)}\, dx$ using the left endpoint. Draw a graph showing the sum.
\soln{9cm}{$\Delta x = \frac{b-a}{n}=\frac{2}{3}$ and $x_i = a+i\Delta x = \frac{2 i}{3} + 1$. Drawing a graph gives:
\begin{center}\begin{tikzpicture}[yscale=2]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-1,-1) grid (6,2.10000000000000);
\draw[latex-latex, very thick] (-1.4,0)--(6.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-1.1)--(0,2.2) node[above]{$y$};
\foreach \x in { -1,1,...,6 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -1,-0.5,...,2.10000000000000 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[thick, color=blue, domain={1.0}:{5.0}, samples=200, samples =350] plot(\x, {ln((\x))});
\draw[fill=blue, opacity=0.3](1.00, 0) rectangle (1.67, 0);
\draw[thick](1.00, 0) rectangle (1.67, 0);
\draw[fill=blue, opacity=0.3](1.67, 0.511) rectangle (2.33, 0);
\draw[thick](1.67, 0.511) rectangle (2.33, 0);
\draw[fill=blue, opacity=0.3](2.33, 0.847) rectangle (3.00, 0);
\draw[thick](2.33, 0.847) rectangle (3.00, 0);
\draw[fill=blue, opacity=0.3](3.00, 1.10) rectangle (3.67, 0);
\draw[thick](3.00, 1.10) rectangle (3.67, 0);
\draw[fill=blue, opacity=0.3](3.67, 1.30) rectangle (4.33, 0);
\draw[thick](3.67, 1.30) rectangle (4.33, 0);
\draw[fill=blue, opacity=0.3](4.33, 1.47) rectangle (5.00, 0);
\draw[thick](4.33, 1.47) rectangle (5.00, 0);
\end{tikzpicture}
\end{center}
The sum is $\sum_{i=0}^{5} \frac{2 \ln{\left(\frac{2 i}{3} + 1 \right)}}{3} = 3.4816$. }
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Use 6 rectangles to approximate <img class="equation_image" title=" \displaystyle \int\limits_{1}^{5} \ln{\left(x \right)}\, dx " src="/equation_images/%20%5Cdisplaystyle%20%5Cint%5Climits_%7B1%7D%5E%7B5%7D%20%5Cln%7B%5Cleft%28x%20%5Cright%29%7D%5C%2C%20dx%20" alt="LaTeX: \displaystyle \int\limits_{1}^{5} \ln{\left(x \right)}\, dx " data-equation-content=" \displaystyle \int\limits_{1}^{5} \ln{\left(x \right)}\, dx " /> using the left endpoint. Draw a graph showing the sum. </p> </p><p> <p> <img class="equation_image" title=" \displaystyle \Delta x = \frac{b-a}{n}=\frac{2}{3} " src="/equation_images/%20%5Cdisplaystyle%20%5CDelta%20x%20%3D%20%5Cfrac%7Bb-a%7D%7Bn%7D%3D%5Cfrac%7B2%7D%7B3%7D%20" alt="LaTeX: \displaystyle \Delta x = \frac{b-a}{n}=\frac{2}{3} " data-equation-content=" \displaystyle \Delta x = \frac{b-a}{n}=\frac{2}{3} " /> and <img class="equation_image" title=" \displaystyle x_i = a+i\Delta x = \frac{2 i}{3} + 1 " src="/equation_images/%20%5Cdisplaystyle%20x_i%20%3D%20a%2Bi%5CDelta%20x%20%3D%20%5Cfrac%7B2%20i%7D%7B3%7D%20%2B%201%20" alt="LaTeX: \displaystyle x_i = a+i\Delta x = \frac{2 i}{3} + 1 " data-equation-content=" \displaystyle x_i = a+i\Delta x = \frac{2 i}{3} + 1 " /> . Drawing a graph gives:
<center><?xml version="1.0" encoding="UTF-8"?>
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="234.632pt" height="201.828pt" viewBox="0 0 234.632 201.828" version="1.1">
<defs>
<g>
<symbol overflow="visible" id="glyph0-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph0-1">
<path style="stroke:none;" d="M 3.328125 -3.015625 C 3.390625 -3.265625 3.625 -4.1875 4.3125 -4.1875 C 4.359375 -4.1875 4.609375 -4.1875 4.8125 -4.0625 C 4.53125 -4 4.34375 -3.765625 4.34375 -3.515625 C 4.34375 -3.359375 4.453125 -3.171875 4.71875 -3.171875 C 4.9375 -3.171875 5.25 -3.34375 5.25 -3.75 C 5.25 -4.265625 4.671875 -4.40625 4.328125 -4.40625 C 3.75 -4.40625 3.40625 -3.875 3.28125 -3.65625 C 3.03125 -4.3125 2.5 -4.40625 2.203125 -4.40625 C 1.171875 -4.40625 0.59375 -3.125 0.59375 -2.875 C 0.59375 -2.765625 0.703125 -2.765625 0.71875 -2.765625 C 0.796875 -2.765625 0.828125 -2.796875 0.84375 -2.875 C 1.1875 -3.9375 1.84375 -4.1875 2.1875 -4.1875 C 2.375 -4.1875 2.71875 -4.09375 2.71875 -3.515625 C 2.71875 -3.203125 2.546875 -2.546875 2.1875 -1.140625 C 2.03125 -0.53125 1.671875 -0.109375 1.234375 -0.109375 C 1.171875 -0.109375 0.953125 -0.109375 0.734375 -0.234375 C 0.984375 -0.296875 1.203125 -0.5 1.203125 -0.78125 C 1.203125 -1.046875 0.984375 -1.125 0.84375 -1.125 C 0.53125 -1.125 0.296875 -0.875 0.296875 -0.546875 C 0.296875 -0.09375 0.78125 0.109375 1.21875 0.109375 C 1.890625 0.109375 2.25 -0.59375 2.265625 -0.640625 C 2.390625 -0.28125 2.75 0.109375 3.34375 0.109375 C 4.375 0.109375 4.9375 -1.171875 4.9375 -1.421875 C 4.9375 -1.53125 4.859375 -1.53125 4.828125 -1.53125 C 4.734375 -1.53125 4.71875 -1.484375 4.6875 -1.421875 C 4.359375 -0.34375 3.6875 -0.109375 3.375 -0.109375 C 2.984375 -0.109375 2.828125 -0.421875 2.828125 -0.765625 C 2.828125 -0.984375 2.875 -1.203125 2.984375 -1.640625 Z M 3.328125 -3.015625 "/>
</symbol>
<symbol overflow="visible" id="glyph0-2">
<path style="stroke:none;" d="M 4.84375 -3.796875 C 4.890625 -3.9375 4.890625 -3.953125 4.890625 -4.03125 C 4.890625 -4.203125 4.75 -4.296875 4.59375 -4.296875 C 4.5 -4.296875 4.34375 -4.234375 4.25 -4.09375 C 4.234375 -4.03125 4.140625 -3.734375 4.109375 -3.546875 C 4.03125 -3.296875 3.96875 -3.015625 3.90625 -2.75 L 3.453125 -0.953125 C 3.421875 -0.8125 2.984375 -0.109375 2.328125 -0.109375 C 1.828125 -0.109375 1.71875 -0.546875 1.71875 -0.921875 C 1.71875 -1.375 1.890625 -2 2.21875 -2.875 C 2.375 -3.28125 2.421875 -3.390625 2.421875 -3.59375 C 2.421875 -4.03125 2.109375 -4.40625 1.609375 -4.40625 C 0.65625 -4.40625 0.296875 -2.953125 0.296875 -2.875 C 0.296875 -2.765625 0.390625 -2.765625 0.40625 -2.765625 C 0.515625 -2.765625 0.515625 -2.796875 0.5625 -2.953125 C 0.84375 -3.890625 1.234375 -4.1875 1.578125 -4.1875 C 1.65625 -4.1875 1.828125 -4.1875 1.828125 -3.875 C 1.828125 -3.625 1.71875 -3.359375 1.65625 -3.171875 C 1.25 -2.109375 1.078125 -1.546875 1.078125 -1.078125 C 1.078125 -0.1875 1.703125 0.109375 2.296875 0.109375 C 2.6875 0.109375 3.015625 -0.0625 3.296875 -0.34375 C 3.171875 0.171875 3.046875 0.671875 2.65625 1.203125 C 2.390625 1.53125 2.015625 1.828125 1.5625 1.828125 C 1.421875 1.828125 0.96875 1.796875 0.796875 1.40625 C 0.953125 1.40625 1.09375 1.40625 1.21875 1.28125 C 1.328125 1.203125 1.421875 1.0625 1.421875 0.875 C 1.421875 0.5625 1.15625 0.53125 1.0625 0.53125 C 0.828125 0.53125 0.5 0.6875 0.5 1.171875 C 0.5 1.671875 0.9375 2.046875 1.5625 2.046875 C 2.578125 2.046875 3.609375 1.140625 3.890625 0.015625 Z M 4.84375 -3.796875 "/>
</symbol>
<symbol overflow="visible" id="glyph0-3">
<path style="stroke:none;" d="M 1.90625 -0.53125 C 1.90625 -0.8125 1.671875 -1.0625 1.390625 -1.0625 C 1.09375 -1.0625 0.859375 -0.8125 0.859375 -0.53125 C 0.859375 -0.234375 1.09375 0 1.390625 0 C 1.671875 0 1.90625 -0.234375 1.90625 -0.53125 Z M 1.90625 -0.53125 "/>
</symbol>
<symbol overflow="visible" id="glyph1-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph1-1">
<path style="stroke:none;" d="M 6.5625 -2.296875 C 6.734375 -2.296875 6.921875 -2.296875 6.921875 -2.5 C 6.921875 -2.6875 6.734375 -2.6875 6.5625 -2.6875 L 1.171875 -2.6875 C 1 -2.6875 0.828125 -2.6875 0.828125 -2.5 C 0.828125 -2.296875 1 -2.296875 1.171875 -2.296875 Z M 6.5625 -2.296875 "/>
</symbol>
<symbol overflow="visible" id="glyph2-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph2-1">
<path style="stroke:none;" d="M 2.9375 -6.375 C 2.9375 -6.625 2.9375 -6.640625 2.703125 -6.640625 C 2.078125 -6 1.203125 -6 0.890625 -6 L 0.890625 -5.6875 C 1.09375 -5.6875 1.671875 -5.6875 2.1875 -5.953125 L 2.1875 -0.78125 C 2.1875 -0.421875 2.15625 -0.3125 1.265625 -0.3125 L 0.953125 -0.3125 L 0.953125 0 C 1.296875 -0.03125 2.15625 -0.03125 2.5625 -0.03125 C 2.953125 -0.03125 3.828125 -0.03125 4.171875 0 L 4.171875 -0.3125 L 3.859375 -0.3125 C 2.953125 -0.3125 2.9375 -0.421875 2.9375 -0.78125 Z M 2.9375 -6.375 "/>
</symbol>
<symbol overflow="visible" id="glyph2-2">
<path style="stroke:none;" d="M 2.890625 -3.515625 C 3.703125 -3.78125 4.28125 -4.46875 4.28125 -5.265625 C 4.28125 -6.078125 3.40625 -6.640625 2.453125 -6.640625 C 1.453125 -6.640625 0.6875 -6.046875 0.6875 -5.28125 C 0.6875 -4.953125 0.90625 -4.765625 1.203125 -4.765625 C 1.5 -4.765625 1.703125 -4.984375 1.703125 -5.28125 C 1.703125 -5.765625 1.234375 -5.765625 1.09375 -5.765625 C 1.390625 -6.265625 2.046875 -6.390625 2.40625 -6.390625 C 2.828125 -6.390625 3.375 -6.171875 3.375 -5.28125 C 3.375 -5.15625 3.34375 -4.578125 3.09375 -4.140625 C 2.796875 -3.65625 2.453125 -3.625 2.203125 -3.625 C 2.125 -3.609375 1.890625 -3.59375 1.8125 -3.59375 C 1.734375 -3.578125 1.671875 -3.5625 1.671875 -3.46875 C 1.671875 -3.359375 1.734375 -3.359375 1.90625 -3.359375 L 2.34375 -3.359375 C 3.15625 -3.359375 3.53125 -2.6875 3.53125 -1.703125 C 3.53125 -0.34375 2.84375 -0.0625 2.40625 -0.0625 C 1.96875 -0.0625 1.21875 -0.234375 0.875 -0.8125 C 1.21875 -0.765625 1.53125 -0.984375 1.53125 -1.359375 C 1.53125 -1.71875 1.265625 -1.921875 0.984375 -1.921875 C 0.734375 -1.921875 0.421875 -1.78125 0.421875 -1.34375 C 0.421875 -0.4375 1.34375 0.21875 2.4375 0.21875 C 3.65625 0.21875 4.5625 -0.6875 4.5625 -1.703125 C 4.5625 -2.515625 3.921875 -3.296875 2.890625 -3.515625 Z M 2.890625 -3.515625 "/>
</symbol>
<symbol overflow="visible" id="glyph2-3">
<path style="stroke:none;" d="M 4.46875 -2 C 4.46875 -3.1875 3.65625 -4.1875 2.578125 -4.1875 C 2.109375 -4.1875 1.671875 -4.03125 1.3125 -3.671875 L 1.3125 -5.625 C 1.515625 -5.5625 1.84375 -5.5 2.15625 -5.5 C 3.390625 -5.5 4.09375 -6.40625 4.09375 -6.53125 C 4.09375 -6.59375 4.0625 -6.640625 3.984375 -6.640625 C 3.984375 -6.640625 3.953125 -6.640625 3.90625 -6.609375 C 3.703125 -6.515625 3.21875 -6.3125 2.546875 -6.3125 C 2.15625 -6.3125 1.6875 -6.390625 1.21875 -6.59375 C 1.140625 -6.625 1.125 -6.625 1.109375 -6.625 C 1 -6.625 1 -6.546875 1 -6.390625 L 1 -3.4375 C 1 -3.265625 1 -3.1875 1.140625 -3.1875 C 1.21875 -3.1875 1.234375 -3.203125 1.28125 -3.265625 C 1.390625 -3.421875 1.75 -3.96875 2.5625 -3.96875 C 3.078125 -3.96875 3.328125 -3.515625 3.40625 -3.328125 C 3.5625 -2.953125 3.59375 -2.578125 3.59375 -2.078125 C 3.59375 -1.71875 3.59375 -1.125 3.34375 -0.703125 C 3.109375 -0.3125 2.734375 -0.0625 2.28125 -0.0625 C 1.5625 -0.0625 0.984375 -0.59375 0.8125 -1.171875 C 0.84375 -1.171875 0.875 -1.15625 0.984375 -1.15625 C 1.3125 -1.15625 1.484375 -1.40625 1.484375 -1.640625 C 1.484375 -1.890625 1.3125 -2.140625 0.984375 -2.140625 C 0.84375 -2.140625 0.5 -2.0625 0.5 -1.609375 C 0.5 -0.75 1.1875 0.21875 2.296875 0.21875 C 3.453125 0.21875 4.46875 -0.734375 4.46875 -2 Z M 4.46875 -2 "/>
</symbol>
<symbol overflow="visible" id="glyph2-4">
<path style="stroke:none;" d="M 4.578125 -3.1875 C 4.578125 -3.984375 4.53125 -4.78125 4.1875 -5.515625 C 3.734375 -6.484375 2.90625 -6.640625 2.5 -6.640625 C 1.890625 -6.640625 1.171875 -6.375 0.75 -5.453125 C 0.4375 -4.765625 0.390625 -3.984375 0.390625 -3.1875 C 0.390625 -2.4375 0.421875 -1.546875 0.84375 -0.78125 C 1.265625 0.015625 2 0.21875 2.484375 0.21875 C 3.015625 0.21875 3.78125 0.015625 4.21875 -0.9375 C 4.53125 -1.625 4.578125 -2.40625 4.578125 -3.1875 Z M 2.484375 0 C 2.09375 0 1.5 -0.25 1.328125 -1.203125 C 1.21875 -1.796875 1.21875 -2.71875 1.21875 -3.3125 C 1.21875 -3.953125 1.21875 -4.609375 1.296875 -5.140625 C 1.484375 -6.328125 2.234375 -6.421875 2.484375 -6.421875 C 2.8125 -6.421875 3.46875 -6.234375 3.65625 -5.25 C 3.765625 -4.6875 3.765625 -3.9375 3.765625 -3.3125 C 3.765625 -2.5625 3.765625 -1.890625 3.65625 -1.25 C 3.5 -0.296875 2.9375 0 2.484375 0 Z M 2.484375 0 "/>
</symbol>
<symbol overflow="visible" id="glyph2-5">
<path style="stroke:none;" d="M 1.265625 -0.765625 L 2.328125 -1.796875 C 3.875 -3.171875 4.46875 -3.703125 4.46875 -4.703125 C 4.46875 -5.84375 3.578125 -6.640625 2.359375 -6.640625 C 1.234375 -6.640625 0.5 -5.71875 0.5 -4.828125 C 0.5 -4.28125 1 -4.28125 1.03125 -4.28125 C 1.203125 -4.28125 1.546875 -4.390625 1.546875 -4.8125 C 1.546875 -5.0625 1.359375 -5.328125 1.015625 -5.328125 C 0.9375 -5.328125 0.921875 -5.328125 0.890625 -5.3125 C 1.109375 -5.96875 1.65625 -6.328125 2.234375 -6.328125 C 3.140625 -6.328125 3.5625 -5.515625 3.5625 -4.703125 C 3.5625 -3.90625 3.078125 -3.125 2.515625 -2.5 L 0.609375 -0.375 C 0.5 -0.265625 0.5 -0.234375 0.5 0 L 4.203125 0 L 4.46875 -1.734375 L 4.234375 -1.734375 C 4.171875 -1.4375 4.109375 -1 4 -0.84375 C 3.9375 -0.765625 3.28125 -0.765625 3.0625 -0.765625 Z M 1.265625 -0.765625 "/>
</symbol>
</g>
</defs>
<g id="surface1">
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,100%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M -28.3455 -56.693812 L 170.080281 -56.693812 M -28.3455 0.0015 L 170.080281 0.0015 M -28.3455 56.692906 L 170.080281 56.692906 M -28.3455 113.388219 L 170.080281 113.388219 M -28.3455 -56.693812 L -28.3455 119.056188 M -0.00175 -56.693812 L -0.00175 119.056188 M 28.345906 -56.693812 L 28.345906 119.056188 M 56.693562 -56.693812 L 56.693562 119.056188 M 85.041219 -56.693812 L 85.041219 119.056188 M 113.388875 -56.693812 L 113.388875 119.056188 M 141.732625 -56.693812 L 141.732625 119.056188 M 170.072469 -56.693812 L 170.072469 119.056188 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -33.947063 0.0015 L 175.681844 0.0015 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 0.597656 138.1875 C 2.296875 138.507812 5.0625 139.464844 6.972656 140.578125 L 6.972656 135.796875 C 5.0625 136.914062 2.296875 137.871094 0.597656 138.1875 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 221.703125 138.1875 C 220.003906 137.871094 217.238281 136.914062 215.328125 135.796875 L 215.328125 140.578125 C 217.238281 139.464844 220.003906 138.507812 221.703125 138.1875 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="225.618" y="140.334"/>
</g>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.00175 -56.6235 L -0.00175 118.985875 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 40.28125 200.550781 C 40.601562 198.851562 41.558594 196.089844 42.675781 194.175781 L 37.890625 194.175781 C 39.007812 196.089844 39.964844 198.851562 40.28125 200.550781 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 40.28125 13.464844 C 39.964844 15.164062 39.007812 17.925781 37.890625 19.839844 L 42.675781 19.839844 C 41.558594 17.925781 40.601562 15.164062 40.28125 13.464844 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-2" x="37.662" y="7.61"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -28.3455 11.337438 L -28.3455 -11.338344 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="5.571" y="159.468"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="13.32" y="159.468"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 28.345906 11.337438 L 28.345906 -11.338344 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="66.138" y="159.468"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 85.041219 11.337438 L 85.041219 -11.338344 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-2" x="122.831" y="159.468"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 141.732625 11.337438 L 141.732625 -11.338344 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="179.524" y="159.468"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 -56.693812 L -5.669719 -56.693812 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="18.364" y="197.677"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="26.112" y="197.677"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 -28.346156 L -5.669719 -28.346156 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="10.615" y="169.331"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-4" x="18.364" y="169.331"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="23.345" y="169.331"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="26.112" y="169.331"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 0.0015 L -5.669719 0.0015 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 28.34525 L -5.669719 28.34525 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-4" x="18.364" y="113.053"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="23.345" y="113.053"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="26.112" y="113.053"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 56.692906 L -5.669719 56.692906 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="26.112" y="84.707"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 85.040563 L -5.669719 85.040563 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="18.364" y="56.36"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-3" x="23.345" y="56.36"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="26.112" y="56.36"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 5.670125 113.388219 L -5.669719 113.388219 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-5" x="26.112" y="28.014"/>
</g>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,100%);stroke-opacity:1;stroke-miterlimit:10;" d="M 28.345906 0.0015 L 28.670125 0.646031 L 28.99825 1.28275 L 29.322469 1.915563 L 29.646687 2.540563 L 29.970906 3.15775 L 30.295125 3.771031 L 30.619344 4.372594 L 30.943562 4.97025 L 31.271687 5.564 L 31.595906 6.149938 L 31.920125 6.728063 L 32.244344 7.302281 L 32.568562 7.872594 L 32.892781 8.435094 L 33.220906 8.993688 L 33.545125 9.544469 L 34.193562 10.630406 L 34.517781 11.165563 L 34.842 11.696813 L 35.170125 12.224156 L 35.494344 12.747594 L 35.818562 13.263219 L 36.142781 13.774938 L 36.467 14.28275 L 36.791219 14.78275 L 37.115437 15.278844 L 37.443562 15.778844 L 37.767781 16.267125 L 38.092 16.7515 L 38.416219 17.231969 L 38.740437 17.708531 L 39.064656 18.181188 L 39.392781 18.649938 L 39.717 19.118688 L 40.041219 19.579625 L 40.689656 20.493688 L 41.013875 20.939 L 41.342 21.388219 L 41.990437 22.271031 L 42.314656 22.708531 L 42.963094 23.575719 L 43.287312 24.0015 L 43.615437 24.423375 L 43.939656 24.84525 L 44.263875 25.263219 L 44.588094 25.677281 L 45.236531 26.497594 L 45.564656 26.903844 L 45.888875 27.302281 L 46.213094 27.704625 L 46.537312 28.099156 L 46.861531 28.497594 L 47.18575 28.884313 L 47.513875 29.278844 L 47.838094 29.665563 L 48.162312 30.044469 L 48.486531 30.427281 L 48.81075 30.806188 L 49.459187 31.556188 L 49.787312 31.927281 L 50.111531 32.294469 L 50.43575 32.65775 L 50.759969 33.028844 L 51.084187 33.388219 L 51.408406 33.743688 L 51.736531 34.103063 L 52.06075 34.458531 L 52.384969 34.806188 L 53.033406 35.509313 L 53.357625 35.856969 L 53.68575 36.196813 L 54.009969 36.540563 L 54.334187 36.880406 L 54.658406 37.216344 L 54.982625 37.544469 L 55.306844 37.888219 L 55.631062 38.22025 L 55.959187 38.544469 L 56.283406 38.880406 L 56.607625 39.204625 L 56.931844 39.536656 L 57.580281 40.177281 L 57.908406 40.497594 L 58.232625 40.814 L 58.556844 41.1265 L 58.881062 41.442906 L 59.205281 41.755406 L 59.5295 42.064 L 59.857625 42.372594 L 60.181844 42.681188 L 60.506062 42.981969 L 61.1545 43.591344 L 61.478719 43.892125 L 61.806844 44.189 L 62.455281 44.78275 L 63.103719 45.368688 L 63.427937 45.65775 L 63.752156 45.950719 L 64.080281 46.239781 L 65.052937 47.09525 L 65.701375 47.65775 L 66.0295 47.939 L 66.677937 48.493688 L 67.650594 49.314 L 67.978719 49.583531 L 68.302937 49.856969 L 68.627156 50.1265 L 69.924031 51.189 L 70.252156 51.450719 L 70.900594 51.974156 L 71.87325 52.747594 L 72.201375 53.0015 L 73.174031 53.763219 L 73.822469 54.263219 L 74.150594 54.513219 L 74.474812 54.759313 L 74.799031 55.009313 L 75.12325 55.255406 L 75.771687 55.739781 L 76.095906 55.985875 L 76.424031 56.224156 L 76.74825 56.466344 L 77.720906 57.181188 L 78.045125 57.415563 L 78.37325 57.649938 L 78.697469 57.888219 L 79.345906 58.349156 L 79.670125 58.587438 L 79.994344 58.814 L 80.322469 59.044469 L 80.646687 59.271031 L 80.970906 59.5015 L 81.943562 60.181188 L 82.267781 60.403844 L 82.595906 60.6265 L 82.920125 60.849156 L 83.244344 61.067906 L 83.568562 61.290563 L 83.892781 61.509313 L 84.217 61.731969 L 84.545125 61.950719 L 84.869344 62.165563 L 85.193562 62.384313 L 85.517781 62.599156 L 85.842 62.810094 L 86.166219 63.028844 L 86.494344 63.243688 L 87.791219 64.087438 L 88.115437 64.294469 L 88.439656 64.505406 L 88.767781 64.712438 L 89.416219 65.1265 L 89.740437 65.329625 L 90.064656 65.536656 L 90.388875 65.739781 L 90.717 65.942906 L 91.365437 66.349156 L 92.338094 66.946813 L 92.666219 67.146031 L 92.990437 67.349156 L 93.638875 67.739781 L 93.963094 67.939 L 94.611531 68.329625 L 94.939656 68.521031 L 95.263875 68.72025 L 95.588094 68.911656 L 95.912312 69.099156 L 96.56075 69.481969 L 96.888875 69.673375 L 97.213094 69.860875 L 97.537312 70.052281 L 97.861531 70.239781 L 98.18575 70.431188 L 98.509969 70.614781 L 98.838094 70.798375 L 99.162312 70.985875 L 99.486531 71.177281 L 100.783406 71.911656 L 101.111531 72.091344 L 101.43575 72.271031 L 102.084187 72.638219 L 102.732625 72.997594 L 103.06075 73.173375 L 103.384969 73.353063 L 103.709187 73.528844 L 104.033406 73.712438 L 104.357625 73.884313 L 104.681844 74.060094 L 105.009969 74.235875 L 105.334187 74.411656 L 105.982625 74.755406 L 106.306844 74.935094 L 106.631062 75.103063 L 106.955281 75.274938 L 107.283406 75.446813 L 107.931844 75.790563 L 108.256062 75.958531 L 108.580281 76.134313 L 108.9045 76.302281 L 109.232625 76.47025 L 109.556844 76.634313 L 110.205281 76.97025 L 110.5295 77.134313 L 110.853719 77.310094 L 111.181844 77.478063 L 111.506062 77.634313 L 112.478719 78.1265 L 112.802937 78.294469 L 113.127156 78.458531 L 113.455281 78.6265 L 113.7795 78.786656 L 114.103719 78.950719 L 115.076375 79.431188 L 115.4045 79.591344 L 115.728719 79.7515 L 116.052937 79.90775 L 116.701375 80.228063 L 117.025594 80.384313 L 117.353719 80.540563 L 118.650594 81.165563 L 118.974812 81.317906 L 119.299031 81.474156 L 119.627156 81.630406 L 120.275594 81.935094 L 120.599812 82.091344 L 120.924031 82.239781 L 121.24825 82.392125 L 121.576375 82.544469 L 121.900594 82.696813 L 122.224812 82.84525 L 122.549031 82.997594 L 122.87325 83.146031 L 123.197469 83.298375 L 123.525594 83.446813 L 125.146687 84.189 L 125.470906 84.333531 L 125.799031 84.481969 L 126.447469 84.771031 L 126.771687 84.919469 L 127.420125 85.208531 L 127.74825 85.353063 L 128.072469 85.493688 L 128.396687 85.642125 L 128.720906 85.78275 L 129.045125 85.927281 L 129.369344 86.067906 L 129.697469 86.212438 L 130.345906 86.493688 L 130.670125 86.638219 L 130.994344 86.774938 L 131.642781 87.056188 L 131.970906 87.196813 L 132.295125 87.337438 L 132.619344 87.474156 L 132.943562 87.614781 L 133.592 87.888219 L 133.920125 88.028844 L 135.541219 88.712438 L 135.869344 88.84525 L 136.517781 89.118688 L 136.842 89.2515 L 137.166219 89.388219 L 137.814656 89.653844 L 138.142781 89.790563 L 138.467 89.919469 L 138.791219 90.056188 L 139.115437 90.185094 L 139.763875 90.450719 L 140.092 90.583531 L 140.416219 90.712438 L 140.740437 90.84525 L 141.064656 90.974156 L 141.388875 91.106969 L 141.713094 91.235875 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 28.345906 0.0015 L 47.338094 0.0015 Z M 28.345906 0.0015 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 28.345906 0.0015 L 47.338094 0.0015 Z M 28.345906 0.0015 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,100%);fill-opacity:0.3;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 47.338094 28.97025 L 47.338094 0.0015 L 66.049031 0.0015 L 66.049031 28.97025 Z M 47.338094 28.97025 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 47.338094 28.97025 L 47.338094 0.0015 L 66.049031 0.0015 L 66.049031 28.97025 Z M 47.338094 28.97025 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,100%);fill-opacity:0.3;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 66.049031 48.021031 L 66.049031 0.0015 L 85.041219 0.0015 L 85.041219 48.021031 Z M 66.049031 48.021031 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 66.049031 48.021031 L 66.049031 0.0015 L 85.041219 0.0015 L 85.041219 48.021031 Z M 66.049031 48.021031 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,100%);fill-opacity:0.3;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 85.041219 62.364781 L 85.041219 0.0015 L 104.033406 0.0015 L 104.033406 62.364781 Z M 85.041219 62.364781 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 85.041219 62.364781 L 85.041219 0.0015 L 104.033406 0.0015 L 104.033406 62.364781 Z M 85.041219 62.364781 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,100%);fill-opacity:0.3;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 104.033406 73.700719 L 104.033406 0.0015 L 122.740437 0.0015 L 122.740437 73.700719 Z M 104.033406 73.700719 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 104.033406 73.700719 L 104.033406 0.0015 L 122.740437 0.0015 L 122.740437 73.700719 Z M 104.033406 73.700719 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,100%);fill-opacity:0.3;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M 122.740437 83.341344 L 122.740437 0.0015 L 141.732625 0.0015 L 141.732625 83.341344 Z M 122.740437 83.341344 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
<path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 122.740437 83.341344 L 122.740437 0.0015 L 141.732625 0.0015 L 141.732625 83.341344 Z M 122.740437 83.341344 " transform="matrix(1,0,0,-1,40.283,138.189)"/>
</g>
</svg>
</center>
The sum is <img class="equation_image" title=" \displaystyle \sum_{i=0}^{5} \frac{2 \ln{\left(\frac{2 i}{3} + 1 \right)}}{3} = 3.4816 " src="/equation_images/%20%5Cdisplaystyle%20%5Csum_%7Bi%3D0%7D%5E%7B5%7D%20%5Cfrac%7B2%20%5Cln%7B%5Cleft%28%5Cfrac%7B2%20i%7D%7B3%7D%20%2B%201%20%5Cright%29%7D%7D%7B3%7D%20%3D%203.4816%20" alt="LaTeX: \displaystyle \sum_{i=0}^{5} \frac{2 \ln{\left(\frac{2 i}{3} + 1 \right)}}{3} = 3.4816 " data-equation-content=" \displaystyle \sum_{i=0}^{5} \frac{2 \ln{\left(\frac{2 i}{3} + 1 \right)}}{3} = 3.4816 " /> . </p> </p>