Problem source

Justify, by means of a sketch, the formula
$$
\lim_{n\rightarrow\infty}\left\{{1\over n}\sum_{m=1}^n
\f(1+m/n)\right\} = \int_1^2 \f(x)\,\d x \,.
$$
Show that 
$$
\lim_{n\rightarrow\infty}\left\{{1\over n+1} + {1\over n+2} + \cdots
+ {1\over n+n}\right\} = \ln 2 \,.
$$
Evaluate
$$
\lim_{n\rightarrow\infty}\left\{{n\over n^2+1} + {n\over n^2+4}
 + \cdots + {n\over n^2+n^2}\right\}\,.
$$

Solution source

\begin{center}
\begin{tikzpicture}[scale=3]
    % Define the function f(x) = 0.4x^2 + 0.5 for visualization
    \def\func(#1){(0.4*(#1)*(#1) + 0.5)}

    % 1. Setup Axes
    \draw[->] (-0.2,0) -- (2.5,0) node[right] {$x$};
    \draw[->] (0,-0.2) -- (0,2.5) node[above] {$y$};

    % 2. Draw the 'Exact' Area (The Integral)
    % We fill the area under the curve lightly
    \fill[gray!10] (1,0) -- plot[domain=1:2, samples=100] (\x,{\func(\x)}) -- (2,0) -- cycle;

    % 3. Draw the Riemann Sum Rectangles
    % We use n=5 for the sketch to make it readable
    \def\n{5}
    \def\dx{1/\n} % Width = 0.2
    
    \foreach \m in {1,...,\n} {
        % Calculate x coordinates
        \pgfmathsetmacro\xright{1 + \m*\dx}       % Right endpoint (1 + m/n)
        \pgfmathsetmacro\xleft{1 + (\m-1)*\dx}    % Left side of rectangle
        \pgfmathsetmacro\height{\func(\xright)}   % Height evaluated at right endpoint
        
        % Draw the rectangle
        \draw[fill=blue!30, fill opacity=0.6, draw=blue!80!black] 
            (\xleft,0) rectangle (\xright,\height);
    }

    % 4. Draw the Function Curve
    \draw[thick, red!80!black] plot[domain=0.5:2.3, samples=100] (\x,{\func(\x)}) node[right] {$y=f(x)$};

    % 5. Annotations and Labels
    
    % Interval boundaries
    \draw[thick] (1,0.05) -- (1,-0.05) node[below] {$1$};
    \draw[thick] (2,0.05) -- (2,-0.05) node[below] {$2$};
    
    % Highlight a specific generic rectangle (e.g., the 3rd one) to label variables
    % Corresponds to m=3
    \pgfmathsetmacro\mgen{3}
    \pgfmathsetmacro\xgen{1+\mgen*\dx} % 1.6
    \pgfmathsetmacro\hgen{\func(\xgen)}
    
    % Dash line to y-axis for height
    \draw[dashed, thin] (\xgen, \hgen) -- (0, \hgen) node[left, font=\footnotesize] {$f(1+\frac{m}{n})$};
    
    % Label the x-point 1+m/n
    \draw (\xgen, 0.05) -- (\xgen, -0.05);
    \node[below, font=\footnotesize, fill=white, inner sep=1pt] at (\xgen, -0.5) {$1+\frac{m}{n}$};
    \draw[->, thin, gray] (\xgen, -0.5) -- (\xgen, -0.05);

    % Label the width 1/n
    \draw[<->, >=stealth] (1.4, 0.4) -- (1.6, 0.4) node[midway, above, font=\footnotesize] {$\frac{1}{n}$};

    % 6. Connecting Text
    \node[align=left, anchor=north west] at (0.2, 2.3) {
        Area $\approx \displaystyle \sum \underbrace{\frac{1}{n}}_{\text{width}} \times \underbrace{f\left(1+\frac{m}{n}\right)}_{\text{height}}$
    };

\end{tikzpicture}
\end{center}


\begin{align*}
&& V &= \lim_{n\rightarrow\infty}\left\{{1\over n+1} + {1\over n+2} + \cdots
+ {1\over n+n}\right\} \\
&& &= \lim_{n\rightarrow\infty}\left\{\sum_{m=1}^n \frac{1}{n+m}\right\} \\
&& &= \lim_{n\rightarrow\infty}\left\{\frac1n\sum_{m=1}^n \frac{1}{1+\frac{m}{n}}\right\} \\
&&&=\int_1^2 \frac{1}{x} \d x \\
&&&= \left [\ln x \right]_1^2 = \ln 2
\end{align*}

\begin{align*}
V &= \lim_{n\rightarrow\infty}\left\{{n\over n^2+1} + {n\over n^2+4}
 + \cdots + {n\over n^2+n^2}\right\} \\
&= \lim_{n\rightarrow\infty}\left\{\sum_{m=1}^n \frac{n}{n^2+m^2}\right\} \\
&= \lim_{n\rightarrow\infty}\left\{\frac{1}{n}\sum_{m=1}^n \frac{1}{1+\left (\frac{m}{n} \right)^2}\right\} \\
&= \int_0^1 \frac{1}{1+x^2} \d x \\
&= \left [\tan^{-1} x  \right]_0^1 \\
&= \frac{\pi}4
\end{align*}