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2026 summer edits #442

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545e54d
Update sec_riemann.ptx
APEXCalculus Jul 1, 2026
ac60c50
Completes height adjustment of marginal figures through Chpt 9, Seque…
APEXCalculus Jul 11, 2026
b20d461
Merge branch '2026-SummerEdits' of https://github.com/APEXCalculus/AP…
APEXCalculus Jul 11, 2026
1bd8ec0
Another edit to Chapter 9
APEXCalculus Jul 11, 2026
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2 changes: 1 addition & 1 deletion ptx/apex.ptx
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Expand Up @@ -83,7 +83,7 @@
<xi:include href="./chapter_app_of_int.ptx"/>

<xi:include href="./chapter_differential_equations.ptx"/>

<xi:include href="./chapter_sequences_series.ptx"/>

<xi:include href="./chapter_planar_curves.ptx"/>
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2 changes: 1 addition & 1 deletion ptx/sec_FTC.ptx
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Expand Up @@ -129,7 +129,7 @@
<m>x \geq 1</m> is given by <m>A(x)=\frac12 (x)(2x)-\frac12 (1)(2)=x^2-1</m>.
</p>

<figure xml:id="fig_ftc1b" vshift="-2">
<figure xml:id="fig_ftc1b" vshift="-1">
<caption>The area of the shaded region is <m>F(x) = \int_1^x 2t\, dt</m></caption>
<image width="47%">
<shortdescription>
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2 changes: 1 addition & 1 deletion ptx/sec_Modeling.ptx
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Expand Up @@ -42,7 +42,7 @@
\draw [firstcolor] (-1.5,0) node [text width=60pt,align=center] (a) { \centering The rate of change of the population};
\draw [firstcolor,-&gt;] (a) -- (.3,.7);

\draw [firstcolor] (2,.25) node [text width=60pt,align=center] (b) { \centering the population.};
\draw [firstcolor] (2,0) node [text width=60pt,align=center] (b) { \centering the\\ population.};
\draw [firstcolor,-&gt;] (b) -- (1.6,.8);

\draw [firstcolor] (.5,2) node [text width=32pt,align=center] (c) { \centering is};
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23 changes: 12 additions & 11 deletions ptx/sec_int_comp_tests.ptx
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Expand Up @@ -513,25 +513,26 @@
Since the limit on the left side diverges to <m>\infty</m>,
we can say that <m>\lim\limits_{n \to \infty}\sum_{i=N}^n b_i</m> also diverges to <m>\infty</m>.
</p>
<aside vshift="2">
<p>
A sequence <m>\{a_n\}</m> is a <em>positive sequence</em>
if <m>a_n \gt 0</m> for all <m>n</m>.
</p>

<p>
Because of <xref ref="thm_series_behavior"/>,
any theorem that relies on a positive sequence still holds true when <m>a_n \gt 0</m> for all but a finite number of values of <m>n</m>.
<idx><h>sequence</h><h>positive</h></idx>
</p>
</aside>
</li>
</ol>
</p>
</proof>

</theorem>

<aside vshift="2">
<p>
A sequence <m>\{a_n\}</m> is a <em>positive sequence</em>
if <m>a_n \gt 0</m> for all <m>n</m>.
</p>

<p>
Because of <xref ref="thm_series_behavior"/>,
any theorem that relies on a positive sequence still holds true when <m>a_n \gt 0</m> for all but a finite number of values of <m>n</m>.
<idx><h>sequence</h><h>positive</h></idx>
</p>
</aside>

<figure xml:id="vid-seqseries-intcomp-comp-test" component="video" vshift="10">
<caption>Video presentation of <xref ref="thm_series_direct_compare"/></caption>
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2 changes: 1 addition & 1 deletion ptx/sec_lhopitals_rule.ptx
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Expand Up @@ -814,7 +814,7 @@
</pg-code>
<statement>
<p>
<m>\lim\limits_{x\to 1} \frac{<var name="$fu"/> }{<var name="$fl"/>}</m>
<md>\lim\limits_{x\to 1} \frac{<var name="$fu"/> }{<var name="$fl"/>}</md>
</p>

<p>
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6 changes: 3 additions & 3 deletions ptx/sec_riemann.ptx
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Expand Up @@ -794,13 +794,13 @@
\draw (1.5,1) node {$\displaystyle x_i = x_0 + i\Delta x$};

\draw [firstcolor] (1,0) node [text width=32pt,align=center] (a) { \centering starting\\[-5pt] value};
\draw [firstcolor,-&gt;] (a) -- (.95,.75);
\draw [firstcolor,-&gt;] (a) -- (1.25,.75);

\draw [firstcolor] (2,2.5) node [text width=120pt,align=center] (b) { \centering number of subintervals\\[-5pt] between $x_0$ and $x_i$};
\draw [firstcolor,-&gt;] (b) -- (1.95,1.25);
\draw [firstcolor,-&gt;] (b) -- (2.05,1.25);

\draw [firstcolor] (3,0) node [text width=60pt,align=center] (c) { \centering subinterval\\[-5pt] size};
\draw [firstcolor,-&gt;] (c) -- (2.75,.75);
\draw [firstcolor,-&gt;] (c) -- (2.45,.75);

\end{tikzpicture}

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182 changes: 91 additions & 91 deletions ptx/sec_sequences.ptx
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Expand Up @@ -733,7 +733,7 @@
</p>

<figure xml:id="fig_seq4c" vshift="-3">
<caption>Scatter plot for the sequence in <xref ref="item_ex_seq4_3"/></caption>
<caption>Scatter plot for the sequence in <xref ref="item_ex_seq4_3"/> of <xref ref="ex_seq4"/></caption>
<!-- START figures/fig_seq4c.tex -->
<image width="47%">
<shortdescription>A scatter plot showing a representative sample of points from the third sequence in this example.</shortdescription>
Expand Down Expand Up @@ -1514,7 +1514,7 @@

<p>
<ol>
<li>
<li xml:id="ex_seq7_1">
<p>
<md>
<mrow>a_{n+1}-a_n \amp = \frac{n+2}{n+1} - \frac{n+1}{n}</mrow>
Expand All @@ -1523,84 +1523,13 @@
<mrow>\amp \lt 0 \text{ for all \(n\). }</mrow>
</md>
Since <m>a_{n+1}-a_n\lt 0</m> for all <m>n</m>,
we conclude that the sequence is decreasing.
</p>
</li>

<li>
<p>
<md>
<mrow>a_{n+1}-a_n \amp = \frac{(n+1)^2+1}{n+2} - \frac{n^2+1}{n+1}</mrow>
<mrow>\amp = \frac{\big((n+1)^2+1\big)(n+1)- (n^2+1)(n+2)}{(n+1)(n+2)}</mrow>
<mrow>\amp = \frac{n^2+3n}{(n+1)(n+2)}</mrow>
<mrow>\amp \gt 0 \text{ for all \(n\). }</mrow>
</md>
Since <m>a_{n+1}-a_n\gt 0</m> for all <m>n</m>,
we conclude the sequence is increasing.
</p>
</li>

<li>
<p>
We can clearly see in <xref ref="fig_seq7c"/>,
where the sequence is plotted, that it is not monotonic.
However, it does seem that after the first 4 terms it is decreasing.
To understand why, perform the same analysis as done before:

<md>
<mrow>a_{n+1}-a_n \amp = \frac{(n+1)^2-9}{(n+1)^2-10(n+1)+26} - \frac{n^2-9}{n^2-10n+26}</mrow>
<mrow>\amp = \frac{n^2+2n-8}{n^2-8n+17}-\frac{n^2-9}{n^2-10n+26}</mrow>
<mrow>\amp = \frac{(n^2+2n-8)(n^2-10n+26)-(n^2-9)(n^2-8n+17)}{(n^2-8n+17)(n^2-10n+26)}</mrow>
<mrow>\amp = \frac{-10n^2+60n-55}{(n^2-8n+17)(n^2-10n+26)}</mrow>
</md>.
we conclude that the sequence is decreasing, illustrated in <xref ref="fig_seq7a"/>.
</p>

<p>
We want to know when this is greater than, or less than, 0.
The denominator is always positive,
therefore we are only concerned with the numerator.
For small values of <m>n</m>,
the numerator is positive.
As <m>n</m> grows large,
the numerator is dominated by <m>-10n^2</m>,
meaning the entire fraction will be negative;
<ie/>, for large enough <m>n</m>, <m>a_{n+1}-a_n \lt 0</m>.
Using the quadratic formula we can determine that the numerator is negative for <m>n\geq 5</m>.
In short, the sequence is simply not monotonic,
though it is useful to note that for <m>n\geq 5</m>,
the sequence is monotonically decreasing.
</p>
</li>

<li xml:id="ex_seq7_4">

<p>
Again, the plot in <xref ref="fig_seq7d"/>
shows that the sequence is not monotonic,
but it suggests that it is monotonically decreasing after the first term.
We perform the usual analysis to confirm this.
<md>
<mrow>a_{n+1}-a_n \amp = \frac{(n+1)^2}{(n+1)!} - \frac{n^2}{n!}</mrow>
<mrow>\amp = \frac{(n+1)^2-n^2(n+1)}{(n+1)!}</mrow>
<mrow>\amp = \frac{-n^3+2n+1}{(n+1)!}</mrow>
</md>
When <m>n=1</m>, the above expression is <m>\gt 0</m>;
for <m>n\geq 2</m>, the above expression is <m>\lt 0</m>.
Thus this sequence is not monotonic,
but it is monotonically decreasing after the first term.
</p>
</li>
</ol>
</p>

<figure xml:id="fig_seq7">
<caption>Plots of sequences in <xref ref="ex_seq7"/></caption>
<sbsgroup>
<sidebyside widths="47% 47%" valign="bottom" margins="0%">
<figure xml:id="fig_seq7a">
<caption/>
<figure xml:id="fig_seq7a" vshift="3">
<caption>Plot of the sequence in <xref ref="ex_seq7_1"/> of <xref ref="ex_seq7"/></caption>
<!-- START figures/fig_seq7a.tex -->
<image>
<image width="47%">
<shortdescription>Plot of the first sequence in this example. It is decreasing and bounded below.</shortdescription>
<description>
<p>
Expand Down Expand Up @@ -1631,11 +1560,24 @@
</image>
<!-- figures/fig_seq7a.tex END -->
</figure>
</li>

<li xml:id="ex_seq7_2">
<p>
<md>
<mrow>a_{n+1}-a_n \amp = \frac{(n+1)^2+1}{n+2} - \frac{n^2+1}{n+1}</mrow>
<mrow>\amp = \frac{\big((n+1)^2+1\big)(n+1)- (n^2+1)(n+2)}{(n+1)(n+2)}</mrow>
<mrow>\amp = \frac{n^2+3n}{(n+1)(n+2)}</mrow>
<mrow>\amp \gt 0 \text{ for all \(n\). }</mrow>
</md>
Since <m>a_{n+1}-a_n\gt 0</m> for all <m>n</m>,
we conclude the sequence is increasing, illustrated in <xref ref="fig_seq7b"/>.
</p>

<figure xml:id="fig_seq7b">
<caption/>
<figure xml:id="fig_seq7b" vshift="2">
<caption>Plot of the sequence in <xref ref="ex_seq7_2"/> of <xref ref="ex_seq7"/></caption>
<!-- START figures/fig_seq7b.tex -->
<image>
<image width="47%">
<shortdescription>Scatter plot for the second sequence in this example. It is increasing but not bounded.</shortdescription>
<description>
<p>
Expand Down Expand Up @@ -1667,13 +1609,43 @@
</image>
<!-- figures/fig_seq7b.tex END -->
</figure>
</sidebyside>
</li>

<li xml:id="ex_seq7_3">
<p>
We can clearly see in <xref ref="fig_seq7c"/>,
where the sequence is plotted, that it is not monotonic.
However, it does seem that after the first 4 terms it is decreasing.
To understand why, perform the same analysis as done before:

<md>
<mrow>a_{n+1}-a_n \amp = \frac{(n+1)^2-9}{(n+1)^2-10(n+1)+26} - \frac{n^2-9}{n^2-10n+26}</mrow>
<mrow>\amp = \frac{n^2+2n-8}{n^2-8n+17}-\frac{n^2-9}{n^2-10n+26}</mrow>
<mrow>\amp = \frac{(n^2+2n-8)(n^2-10n+26)-(n^2-9)(n^2-8n+17)}{(n^2-8n+17)(n^2-10n+26)}</mrow>
<mrow>\amp = \frac{-10n^2+60n-55}{(n^2-8n+17)(n^2-10n+26)}</mrow>
</md>.
</p>

<sidebyside widths="47% 47%" valign="bottom" margins="0%">
<figure xml:id="fig_seq7c">
<caption/>
<p>
We want to know when this is greater than, or less than, 0.
The denominator is always positive,
therefore we are only concerned with the numerator.
For small values of <m>n</m>,
the numerator is positive.
As <m>n</m> grows large,
the numerator is dominated by <m>-10n^2</m>,
meaning the entire fraction will be negative;
<ie/>, for large enough <m>n</m>, <m>a_{n+1}-a_n \lt 0</m>.
Using the quadratic formula we can determine that the numerator is negative for <m>n\geq 5</m>.
In short, the sequence is simply not monotonic,
though it is useful to note that for <m>n\geq 5</m>,
the sequence is monotonically decreasing.
</p>

<figure xml:id="fig_seq7c" vshift="3">
<caption>Plot of the sequence in <xref ref="ex_seq7_3"/> of <xref ref="ex_seq7"/></caption>
<!-- START figures/fig_seq7c.tex -->
<image>
<image width="47%">
<shortdescription>Scatter plot for the third sequence in this example. It is not monotonic.</shortdescription>
<description>
<p>
Expand Down Expand Up @@ -1710,12 +1682,32 @@
<!-- figures/fig_seq7c.tex END -->
</figure>

<figure xml:id="fig_seq7d">
<caption/>
</li>

<li xml:id="ex_seq7_4">

<p>
Again, the plot in <xref ref="ex_seq7_4"/> of <xref ref="fig_seq7d"/>
shows that the sequence is not monotonic,
but it suggests that it is monotonically decreasing after the first term.
We perform the usual analysis to confirm this.
<md>
<mrow>a_{n+1}-a_n \amp = \frac{(n+1)^2}{(n+1)!} - \frac{n^2}{n!}</mrow>
<mrow>\amp = \frac{(n+1)^2-n^2(n+1)}{(n+1)!}</mrow>
<mrow>\amp = \frac{-n^3+2n+1}{(n+1)!}</mrow>
</md>
When <m>n=1</m>, the above expression is <m>\gt 0</m>;
for <m>n\geq 2</m>, the above expression is <m>\lt 0</m>.
Thus this sequence is not monotonic,
but it is monotonically decreasing after the first term.
</p>

<figure xml:id="fig_seq7d" vshift="3">
<caption>Plot of the sequence in <xref ref="ex_seq7_4"/> of <xref ref="ex_seq7"/></caption>
<!-- <caption>A plot of <m>\{a_n\} = \{n^2/n!\}</m> in <xref ref="ex_seq7_4"/></caption> -->
<!-- Either all get captions, or none. The labels in each image make it clear enough?-->
<!-- START figures/fig_seq7d.tex -->
<image>
<image width="47%">
<shortdescription>Scatter plot for the last sequence in this example. It is not monotonic.</shortdescription>
<description>
<p>
Expand Down Expand Up @@ -1754,9 +1746,17 @@
</image>
<!-- figures/fig_seq7d.tex END -->
</figure>
</sidebyside>
</sbsgroup>
</figure>

</li>
</ol>
</p>







</solution>
<solution component="video" vshift="0">
<title>Video solution</title>
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