\begin{patch}{ExSumGeneralFunctionPatch1}
\begin{paste}{ExSumGeneralFunctionFull1}{ExSumGeneralFunctionEmpty1}
\pastebutton{ExSumGeneralFunctionFull1}{\hidepaste}
\tab{5}\spadcommand{sum(k * x^k,k = 1..n)}
\indentrel{3}\begin{verbatim}
            2               n
        (n x  + (- n - 1)x)x  + x
   (1)  -------------------------
                2
               x  - 2 x + 1
                              Type: Expression(Integer)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumGeneralFunctionEmpty1}
\begin{paste}{ExSumGeneralFunctionEmpty1}{ExSumGeneralFunctionPatch1}
\pastebutton{ExSumGeneralFunctionEmpty1}{\showpaste}
\tab{5}\spadcommand{sum(k * x^k,k = 1..n)}
\end{paste}\end{patch}

\begin{patch}{ExSumInfinitePatch1}
\begin{paste}{ExSumInfiniteFull1}{ExSumInfiniteEmpty1}
\pastebutton{ExSumInfiniteFull1}{\hidepaste}
\tab{5}\spadcommand{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)}
\indentrel{3}\begin{verbatim}
        3
   (1)  -
        4
Type: Union(OrderedCompletion(Fraction(Polynomial(Integer))),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumInfiniteEmpty1}
\begin{paste}{ExSumInfiniteEmpty1}{ExSumInfinitePatch1}
\pastebutton{ExSumInfiniteEmpty1}{\showpaste}
\tab{5}\spadcommand{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)}
\end{paste}\end{patch}

\begin{patch}{ExSumClosedFormPatch1}
\begin{paste}{ExSumClosedFormFull1}{ExSumClosedFormEmpty1}
\pastebutton{ExSumClosedFormFull1}{\hidepaste}
\tab{5}\spadcommand{s := sum(k^2,k = a..b)\bound{s }}
\indentrel{3}\begin{verbatim}
           3      2          3      2
        2 b  + 3 b  + b - 2 a  + 3 a  - a
   (1)  ---------------------------------
                        6
                    Type: Fraction(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumClosedFormEmpty1}
\begin{paste}{ExSumClosedFormEmpty1}{ExSumClosedFormPatch1}
\pastebutton{ExSumClosedFormEmpty1}{\showpaste}
\tab{5}\spadcommand{s := sum(k^2,k = a..b)\bound{s }}
\end{paste}\end{patch}

\begin{patch}{ExSumClosedFormPatch2}
\begin{paste}{ExSumClosedFormFull2}{ExSumClosedFormEmpty2}
\pastebutton{ExSumClosedFormFull2}{\hidepaste}
\tab{5}\spadcommand{eval(s,[a,b],[1,25])\free{s }}
\indentrel{3}\begin{verbatim}
   (2)  5525
                    Type: Fraction(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumClosedFormEmpty2}
\begin{paste}{ExSumClosedFormEmpty2}{ExSumClosedFormPatch2}
\pastebutton{ExSumClosedFormEmpty2}{\showpaste}
\tab{5}\spadcommand{eval(s,[a,b],[1,25])\free{s }}
\end{paste}\end{patch}

\begin{patch}{ExSumClosedFormPatch3}
\begin{paste}{ExSumClosedFormFull3}{ExSumClosedFormEmpty3}
\pastebutton{ExSumClosedFormFull3}{\hidepaste}
\tab{5}\spadcommand{reduce(+,[i^2 for i in 1..25])}
\indentrel{3}\begin{verbatim}
   (3)  5525
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumClosedFormEmpty3}
\begin{paste}{ExSumClosedFormEmpty3}{ExSumClosedFormPatch3}
\pastebutton{ExSumClosedFormEmpty3}{\showpaste}
\tab{5}\spadcommand{reduce(+,[i^2 for i in 1..25])}
\end{paste}\end{patch}

\begin{patch}{ExSumPolynomialPatch1}
\begin{paste}{ExSumPolynomialFull1}{ExSumPolynomialEmpty1}
\pastebutton{ExSumPolynomialFull1}{\hidepaste}
\tab{5}\spadcommand{sum(3*k^2/(c^2 + 1) + 12*k/d,k = (3*a)..(4*b))}
\indentrel{3}\begin{verbatim}
   (1)
             3       2             3       2
       (128 b  + 48 b  + 4 b - 54 a  + 27 a  - 3 a)d
     + 
             2               2         2        2
       (192 b  + 48 b - 108 a  + 36 a)c  + 192 b  + 48 b
     + 
              2
       - 108 a  + 36 a
  /
         2
     (2 c  + 2)d
         Type: Union(Fraction(Polynomial(Integer)),...)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumPolynomialEmpty1}
\begin{paste}{ExSumPolynomialEmpty1}{ExSumPolynomialPatch1}
\pastebutton{ExSumPolynomialEmpty1}{\showpaste}
\tab{5}\spadcommand{sum(3*k^2/(c^2 + 1) + 12*k/d,k = (3*a)..(4*b))}
\end{paste}\end{patch}

\begin{patch}{ExSumListEntriesIPatch1}
\begin{paste}{ExSumListEntriesIFull1}{ExSumListEntriesIEmpty1}
\pastebutton{ExSumListEntriesIFull1}{\hidepaste}
\tab{5}\spadcommand{[i for i in 1..15]}
\indentrel{3}\begin{verbatim}
   (1)
   [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]
                            Type: List(PositiveInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumListEntriesIEmpty1}
\begin{paste}{ExSumListEntriesIEmpty1}{ExSumListEntriesIPatch1}
\pastebutton{ExSumListEntriesIEmpty1}{\showpaste}
\tab{5}\spadcommand{[i for i in 1..15]}
\end{paste}\end{patch}

\begin{patch}{ExSumListEntriesIPatch2}
\begin{paste}{ExSumListEntriesIFull2}{ExSumListEntriesIEmpty2}
\pastebutton{ExSumListEntriesIFull2}{\hidepaste}
\tab{5}\spadcommand{reduce(+,[i for i in 1..15])}
\indentrel{3}\begin{verbatim}
   (2)  120
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumListEntriesIEmpty2}
\begin{paste}{ExSumListEntriesIEmpty2}{ExSumListEntriesIPatch2}
\pastebutton{ExSumListEntriesIEmpty2}{\showpaste}
\tab{5}\spadcommand{reduce(+,[i for i in 1..15])}
\end{paste}\end{patch}

\begin{patch}{ExSumApproximateEPatch1}
\begin{paste}{ExSumApproximateEFull1}{ExSumApproximateEEmpty1}
\pastebutton{ExSumApproximateEFull1}{\hidepaste}
\tab{5}\spadcommand{reduce(+,[1.0/factorial(n) for n in 0..20])}
\indentrel{3}\begin{verbatim}
   (1)  2.7182818284_590452354
                                            Type: Float
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumApproximateEEmpty1}
\begin{paste}{ExSumApproximateEEmpty1}{ExSumApproximateEPatch1}
\pastebutton{ExSumApproximateEEmpty1}{\showpaste}
\tab{5}\spadcommand{reduce(+,[1.0/factorial(n) for n in 0..20])}
\end{paste}\end{patch}

\begin{patch}{ExSumListEntriesIIPatch1}
\begin{paste}{ExSumListEntriesIIFull1}{ExSumListEntriesIIEmpty1}
\pastebutton{ExSumListEntriesIIFull1}{\hidepaste}
\tab{5}\spadcommand{[n^2 for n in 5..20]}
\indentrel{3}\begin{verbatim}
   (1)
   [25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225,
    256, 289, 324, 361, 400]
                            Type: List(PositiveInteger)
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumListEntriesIIEmpty1}
\begin{paste}{ExSumListEntriesIIEmpty1}{ExSumListEntriesIIPatch1}
\pastebutton{ExSumListEntriesIIEmpty1}{\showpaste}
\tab{5}\spadcommand{[n^2 for n in 5..20]}
\end{paste}\end{patch}

\begin{patch}{ExSumListEntriesIIPatch2}
\begin{paste}{ExSumListEntriesIIFull2}{ExSumListEntriesIIEmpty2}
\pastebutton{ExSumListEntriesIIFull2}{\hidepaste}
\tab{5}\spadcommand{reduce(+,[n^2 for n in 5..20])}
\indentrel{3}\begin{verbatim}
   (2)  2840
                                  Type: PositiveInteger
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumListEntriesIIEmpty2}
\begin{paste}{ExSumListEntriesIIEmpty2}{ExSumListEntriesIIPatch2}
\pastebutton{ExSumListEntriesIIEmpty2}{\showpaste}
\tab{5}\spadcommand{reduce(+,[n^2 for n in 5..20])}
\end{paste}\end{patch}

\begin{patch}{ExSumCubesPatch1}
\begin{paste}{ExSumCubesFull1}{ExSumCubesEmpty1}
\pastebutton{ExSumCubesFull1}{\hidepaste}
\tab{5}\spadcommand{sum(k^3,k = 1..n)}
\indentrel{3}\begin{verbatim}
         4      3    2
        n  + 2 n  + n
   (1)  --------------
               4
                    Type: Fraction(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumCubesEmpty1}
\begin{paste}{ExSumCubesEmpty1}{ExSumCubesPatch1}
\pastebutton{ExSumCubesEmpty1}{\showpaste}
\tab{5}\spadcommand{sum(k^3,k = 1..n)}
\end{paste}\end{patch}

\begin{patch}{ExSumCubesPatch2}
\begin{paste}{ExSumCubesFull2}{ExSumCubesEmpty2}
\pastebutton{ExSumCubesFull2}{\hidepaste}
\tab{5}\spadcommand{sum(k,k = 1..n) ^ 2}
\indentrel{3}\begin{verbatim}
         4      3    2
        n  + 2 n  + n
   (2)  --------------
               4
                    Type: Fraction(Polynomial(Integer))
\end{verbatim}
\indentrel{-3}\end{paste}\end{patch}

\begin{patch}{ExSumCubesEmpty2}
\begin{paste}{ExSumCubesEmpty2}{ExSumCubesPatch2}
\pastebutton{ExSumCubesEmpty2}{\showpaste}
\tab{5}\spadcommand{sum(k,k = 1..n) ^ 2}
\end{paste}\end{patch}

