quick swing at redefining series

source/calculus/source/08-SQ/02.ptx

@@ -317,7 +317,11 @@
     <definition xml:id="definitionlimitsequence">
         <statement>
         <p>
-            Given a sequence <m>\{x_n\}</m>, we say <m>x_n</m> has <em>limit</em> <m>L</m>, denoted <me>\lim_{n\to\infty} x_n=L</me> if we can make <m>x_n</m> as close to <m>L</m> as we like by making <m>n</m> sufficiently large.  If such an <m>L</m> exists, we say <em><m>\{x_n\}</m> converges to <m>L</m>.</em>  If no such <m>L</m> exists, we say <em><m>\{x_n\}</m> does not converge.</em>
+            Given a sequence <m>\{x_n\}</m>, we say <m>x_n</m> has <term>limit</term> <m>L</m>,
+            denoted <me>\lim_{n\to\infty} x_n=L</me> if we can make <m>x_n</m> as close to <m>L</m>
+            as we like by making <m>n</m> sufficiently large.  If such an <m>L</m> exists, we say
+            <em><m>\{x_n\}</m> <term>converges</term> to <m>L</m>.</em> 
+            If no such <m>L</m> exists, we say <m>\{x_n\}</m> <term>diverges</term>.
         </p>
             </statement>
     </definition>

source/calculus/source/08-SQ/03.ptx

@@ -1,7 +1,7 @@
 <?xml version='1.0' encoding='utf-8'?>
 
 <section xml:id="SQ3" xmlns:xi="http://www.w3.org/2001/XInclude" tbil-slides="build">
-  <title>Partial Sum Sequence (SQ3)</title>
+  <title>Partial Sums and Series (SQ3)</title>
     <objectives>
         <ul>
             <li>
@@ -66,7 +66,6 @@
                 <p>
 Given a sequence <m>\{a_n\}_{n=0}^\infty</m> define the <term><m>k^{\text{th}}</m> partial sum</term> for this sequence to be
 <me>A_k=\sum_{i=0}^k a_i=a_0+a_1+a_2+\cdots+a_k.</me> 
-Note that <m>\{A_n\}_{n=0}^\infty=A_0, A_1, A_2, \ldots</m> is itself a sequence called the <em>partial sum sequence</em>.
                 </p>
                 <p>
 More generally, partial sums may be defined for any starting index. Given <m>\{a_n\}_{n=N}^\infty</m>, let
@@ -251,7 +250,21 @@ More generally, partial sums may be defined for any starting index. Given <m>\{a
         <definition xml:id="definition-SQ3serieslimit">
             <statement>
                 <p>
-                    Given a sequence <m>a_n</m>, we define the limit of the series <me>\displaystyle\sum_{n=k}^\infty a_n:=\lim_{n\to \infty} A_n</me> where <m>A_n=\displaystyle \sum_{i=k}^n a_i</m>.  We call <m>\displaystyle\sum_{n=k}^\infty a_n</m> an <em>infinite series</em>.
+Given a sequence <m>\{a_n\}_{n=k}^\infty</m>, we define its <term>infinite series</term>
+(or just <term>series</term>) to be its sequence of its partial sums
+<me>\left\{A_n\right\}_{n=k}^\infty=\left\{\sum_{i=k}^n a_i\right\}_{n=k}^\infty
+=\left\{a_k,a_k+a_{k+1},a_k+a_{k+1}+a_{k+2},\dots\right\}</me>
+and often use the notation
+<me>\sum_{i=k}^\infty a_i = a_k+a_{k+1}+a_{k+2}+\dots</me>
+to represent it. We will also write <m>\sum a_i</m> for short when the starting index <m>n=k</m>
+is either known from context or irrelevant.
+                </p>
+                <p>
+When the series (the sequence of partial sums) converges to a limit, we say the series is <term>convergent</term> and
+this limit is the <term>value</term> of the series, and write:
+<me>\sum_{i=k}^\infty a_i = a_k+a_{k+1}+a_{k+2}+\dots =
+\lim_{n\to\infty} \sum_{i=k}^n a_i</me>.
+When the series (the sequence of partial sums) diverges, we say the series is <term>divergent</term>.
                 </p>
             </statement>
         </definition>

source/calculus/source/08-SQ/outcomes/03.ptx

@@ -1,4 +1,5 @@
 <?xml version='1.0' encoding='UTF-8'?>
 <p>
-Compute the first few terms of a telescoping or geometric partial sum sequence, and find a closed form for this sequence, and compute its limit.
-</p>
+Compute the first few terms of a telescoping or geometric series,
+find a closed form for this sequence, and compute its limit.
+</p>