<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">%The following command typeset the series of the second
%column of the page 9
%
%This command has one parameter:
%        1) The width of the mathematical text
\newcommand\TNineSeriesThree[1]{%
   %This command typeset a fact about a serie and
   %the math used to describe the property.
   %
   %The command has two parameters:
   %      1) The title of the fact
   %      2) The corresponding math.
   %This macro uses the global (to the macro) variables '\Hspace', '\CurrentLineWidth'
   %which should contain the space used to typeset the math.
   \def\SerieProperty##1##2{%
         \def\temp{\TNineTitle{##1} \ensuremath{##2}}%
         \savebox\TmpBoxA{\temp}%
         \settowidth\CurrentLineWidth{\usebox\TmpBoxA}%
         \ifdimcomp{\CurrentLineWidth}{&lt;}{\HSpace}%
         {%Enough space, everything is written on one line
          \temp
         }{%The title is on one line, the math typeset as 'display math'
           \TNineTitle{##1}%
           \begin{displaymath}%
               ##2%
           \end{displaymath}%
          }%
         \par
         \AdjustSpace{\TNineInterTitle}%
   }%
   %This command typeset a line showing the expansion of ordinary power serie
   %
   %The command has 2 parameters
   %      1) The formula to be expanded
   %      2) The expanded version
   \def\LineOfOrdPowSerie##1##2{##1&amp;=&amp;##2\\[\TNineExpansionSkip]}%
   \parbox[t]{#1}{%
      \TNineSeriesFontSize
      \DisplaySpace{\TNineDisplaySpace}{\TNineDisplayShortSpace}%
      \setlength{\HSpace}{#1}%
      \SerieProperty{Ordinary power series:}
                    {A(x) = \sum_{i=0}^\infty a_i x^i}
       \SerieProperty{Exponential power series:}
                     {A(x) = \sum_{i=0}^\infty a_i \frac{x^i}{i!}}
       \SerieProperty{Dirichlet power series:}
                     {A(x) = \sum_{i=1}^\infty \frac{a_i}{i^x}}
       \SerieProperty{Binomial theorem:}
                     {(x + y)^n = \sum_{k=0}^n\binom{n}{k} x^{n-k} y^k}
       \SerieProperty{Difference of like powers:}
                     {x^n - y^n = (x-y) \sum_{k=0}^{n-1} x^{n-1-k} y^k}
       \SerieProperty{For ordinary power series:}
                    {\begin{array}{lcl}
                       \LineOfOrdPowSerie{\alpha A(x) + \beta B(x)}%
                                         {\sum_{i=0}^\infty(\alpha a_i + \beta b_i) x^i}
                       \LineOfOrdPowSerie{x^k A(x)}%
                                         {\sum_{i=k}^\infty a_{i-k} x^i}
                       \LineOfOrdPowSerie{\frac{A(x)  - \sum_{i=0}^{k-1} a_i x^i}{x^k}}%
                                         {\sum_{i=0}^\infty a_{i+k} x^i}
                       \LineOfOrdPowSerie{A(cx)}%
                                         {\sum_{i=0}^\infty c^i a_i x^i}
                       \LineOfOrdPowSerie{A'(x)}%
                                         {\sum_{i=0}^\infty (i+1) a_{i+1} x^i}
                       \LineOfOrdPowSerie{x A'(x)}%
                                         {\sum_{i=1}^\infty i a_i x^i}
                       \LineOfOrdPowSerie{\int A(x) \, dx}%
                                         {\sum_{i=1}^\infty \frac{a_{i-1}}{i} x^i}
                       \LineOfOrdPowSerie{\frac{A(x) + A(-x)}{2}}%
                                         {\sum_{i=0}^\infty a_{2i} x^{2i}}
                       \LineOfOrdPowSerie{\frac{A(x) - A(-x)}{2}}%
                                         {\sum_{i=0}^\infty a_{2i+1} x^{2i+1}}
                       \end{array}
                      }
       \SerieProperty{Summation:} 
                     {\text{If }b_i = \sum_{j=0}^i a_i\text{ then }
                      B(x) = \frac{1}{1 -x} A(x)}
        \SerieProperty{Convolution:}
                     {A(x)B(x) = \sum_{i=0}^\infty\left(\sum_{j=0}^i a_j b_{i-j}\right)x^i}
   }
}
</pre></body></html>