\begin{patch}{ExSeriesSubstitutionPatch1} \begin{paste}{ExSeriesSubstitutionFull1}{ExSeriesSubstitutionEmpty1} \pastebutton{ExSeriesSubstitutionFull1}{\hidepaste} \tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }} \indentrel{3}\begin{verbatim} (1) 1 2 1 3 1 4 1 5 1 6 1 + x + � x + � x + �� x + ��� x + ��� x 2 6 24 120 720 + 1 7 1 8 1 9 1 10 11 ���� x + ����� x + ������ x + ������� x + O(x ) 5040 40320 362880 3628800 Type: UnivariateTaylorSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesSubstitutionEmpty1} \begin{paste}{ExSeriesSubstitutionEmpty1}{ExSeriesSubstitutionPatch1} \pastebutton{ExSeriesSubstitutionEmpty1}{\showpaste} \tab{5}\spadcommand{f := taylor(exp(x))\bound{f2 }} \end{paste}\end{patch} \begin{patch}{ExSeriesSubstitutionPatch2} \begin{paste}{ExSeriesSubstitutionFull2}{ExSeriesSubstitutionEmpty2} \pastebutton{ExSeriesSubstitutionFull2}{\hidepaste} \tab{5}\spadcommand{eval(f,1.0)\free{f2 }} \indentrel{3}\begin{verbatim} (2) [1.0, 2.0, 2.5, 2.6666666666 666666667, 2.7083333333 333333333, 2.7166666666 666666667, 2.7180555555 555555556, 2.7182539682 53968254, 2.7182787698 412698413, 2.7182815255 731922399, ...] Type: Stream Expression Float \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesSubstitutionEmpty2} \begin{paste}{ExSeriesSubstitutionEmpty2}{ExSeriesSubstitutionPatch2} \pastebutton{ExSeriesSubstitutionEmpty2}{\showpaste} \tab{5}\spadcommand{eval(f,1.0)\free{f2 }} \end{paste}\end{patch} \begin{patch}{ExSeriesConvertPatch1} \begin{paste}{ExSeriesConvertFull1}{ExSeriesConvertEmpty1} \pastebutton{ExSeriesConvertFull1}{\hidepaste} \tab{5}\spadcommand{series(sin(a*x),x = 0)} \indentrel{3}\begin{verbatim} (1) 3 5 7 9 a 3 a 5 a 7 a 9 a x - �� x + ��� x - ���� x + ������ x 6 120 5040 362880 + 11 a 11 12 - �������� x + O(x ) 39916800 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesConvertEmpty1} \begin{paste}{ExSeriesConvertEmpty1}{ExSeriesConvertPatch1} \pastebutton{ExSeriesConvertEmpty1}{\showpaste} \tab{5}\spadcommand{series(sin(a*x),x = 0)} \end{paste}\end{patch} \begin{patch}{ExSeriesConvertPatch2} \begin{paste}{ExSeriesConvertFull2}{ExSeriesConvertEmpty2} \pastebutton{ExSeriesConvertFull2}{\hidepaste} \tab{5}\spadcommand{series(sin(a*x),a = \%pi/4)} \indentrel{3}\begin{verbatim} (2) %pi x %pi x %pi sin(�����) + x cos(�����)(a - ���) 4 4 4 + 2 %pi x 3 %pi x x sin(�����) x cos(�����) 4 %pi 2 4 %pi 3 - ������������ (a - ���) - ������������ (a - ���) 2 4 6 4 + 4 %pi x 5 %pi x x sin(�����) x cos(�����) 4 %pi 4 4 %pi 5 ������������ (a - ���) + ������������ (a - ���) 24 4 120 4 + 6 %pi x 7 %pi x x sin(�����) x cos(�����) 4 %pi 6 4 %pi 7 - ������������ (a - ���) - ������������ (a - ���) 720 4 5040 4 + 8 %pi x 9 %pi x x sin(�����) x cos(�����) 4 %pi 8 4 %pi 9 ������������ (a - ���) + ������������ (a - ���) 40320 4 362880 4 + 10 %pi x x sin(�����) 4 %pi 10 %pi 11 - ������������� (a - ���) + O((a - ���) ) 3628800 4 4 Type: UnivariatePuiseuxSeries(Expression Integer,a,pi/4) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesConvertEmpty2} \begin{paste}{ExSeriesConvertEmpty2}{ExSeriesConvertPatch2} \pastebutton{ExSeriesConvertEmpty2}{\showpaste} \tab{5}\spadcommand{series(sin(a*x),a = \%pi/4)} \end{paste}\end{patch} \begin{patch}{ExSeriesFunctionsPatch1} \begin{paste}{ExSeriesFunctionsFull1}{ExSeriesFunctionsEmpty1} \pastebutton{ExSeriesFunctionsFull1}{\hidepaste} \tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }} \indentrel{3}\begin{verbatim} (1) 2 3 4 5 6 7 8 9 10 1 + x + x + x + x + x + x + x + x + x + x + 11 O(x ) Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesFunctionsEmpty1} \begin{paste}{ExSeriesFunctionsEmpty1}{ExSeriesFunctionsPatch1} \pastebutton{ExSeriesFunctionsEmpty1}{\showpaste} \tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f1 }} \end{paste}\end{patch} \begin{patch}{ExSeriesFunctionsPatch2} \begin{paste}{ExSeriesFunctionsFull2}{ExSeriesFunctionsEmpty2} \pastebutton{ExSeriesFunctionsFull2}{\hidepaste} \tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }} \indentrel{3}\begin{verbatim} (2) 1 2 1 3 1 4 1 5 1 6 1 7 1 8 x + � x + � x + � x + � x + � x + � x + � x 2 3 4 5 6 7 8 + 1 9 1 10 1 11 12 � x + �� x + �� x + O(x ) 9 10 11 Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesFunctionsEmpty2} \begin{paste}{ExSeriesFunctionsEmpty2}{ExSeriesFunctionsPatch2} \pastebutton{ExSeriesFunctionsEmpty2}{\showpaste} \tab{5}\spadcommand{g := log(f)\free{f1 }\bound{g }} \end{paste}\end{patch} \begin{patch}{ExSeriesFunctionsPatch3} \begin{paste}{ExSeriesFunctionsFull3}{ExSeriesFunctionsEmpty3} \pastebutton{ExSeriesFunctionsFull3}{\hidepaste} \tab{5}\spadcommand{exp(g)\free{g }} \indentrel{3}\begin{verbatim} (3) 2 3 4 5 6 7 8 9 10 1 + x + x + x + x + x + x + x + x + x + x + 11 O(x ) Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesFunctionsEmpty3} \begin{paste}{ExSeriesFunctionsEmpty3}{ExSeriesFunctionsPatch3} \pastebutton{ExSeriesFunctionsEmpty3}{\showpaste} \tab{5}\spadcommand{exp(g)\free{g }} \end{paste}\end{patch} \begin{patch}{ExSeriesManipulatePatch1} \begin{paste}{ExSeriesManipulateFull1}{ExSeriesManipulateEmpty1} \pastebutton{ExSeriesManipulateFull1}{\hidepaste} \tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }} \indentrel{3}\begin{verbatim} (1) 2 3 4 5 6 7 8 9 10 1 + x + x + x + x + x + x + x + x + x + x + 11 O(x ) Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesManipulateEmpty1} \begin{paste}{ExSeriesManipulateEmpty1}{ExSeriesManipulatePatch1} \pastebutton{ExSeriesManipulateEmpty1}{\showpaste} \tab{5}\spadcommand{f := series(1/(1-x),x = 0)\bound{f }} \end{paste}\end{patch} \begin{patch}{ExSeriesManipulatePatch2} \begin{paste}{ExSeriesManipulateFull2}{ExSeriesManipulateEmpty2} \pastebutton{ExSeriesManipulateFull2}{\hidepaste} \tab{5}\spadcommand{f ** 2\free{f }} \indentrel{3}\begin{verbatim} (2) 2 3 4 5 6 7 8 1 + 2x + 3x + 4x + 5x + 6x + 7x + 8x + 9x + 9 10 11 10x + 11x + O(x ) Type: UnivariatePuiseuxSeries(Expression Integer,x,0) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{ExSeriesManipulateEmpty2} \begin{paste}{ExSeriesManipulateEmpty2}{ExSeriesManipulatePatch2} \pastebutton{ExSeriesManipulateEmpty2}{\showpaste} \tab{5}\spadcommand{f ** 2\free{f }} \end{paste}\end{patch}