%This command typeset the series of the first column
%of the page 9
%
%The command has one parameter:
%       1) The width usable to type the text
\newcommand\TNineSeriesOne[1]{%
   %The following command typeset the development of the different
   %series.
   %The macro has 3 parameters
   %      1) The closed form (before the first equal sign)
   %      2) The development(between both equal sign)
   %      3) The form written with the sum sign (Σ) after
   %         the third equal sign
   %This macro use the global (to this macro) length
   %'\HSpace' which should have been set to the width
   %of the mathematical text.
   \def\DevelopSerie##1##2##3{##1&=&##2&=&##3\\[\TNineExpansionSkip]}%
   \parbox[t]{#1}{%
      \TNineSeriesFontSize
      \DisplaySpace{\TNineDisplaySpace}{\TNineDisplayShortSpace}%
      \TNineTitle{Taylor's series centered at $a$:}%
      \begin{displaymath}
         f(x) = f(a) + (x-a)f'(a) + \tfrac{(x-a)^2}{2} f''(a) + \cdots = 
         \sum_{i=0}^\infty \tfrac{(x-a)^i}{i!} f^{(i)}(a)
      \end{displaymath}
      \TNineTitle{Expansions:}
      \AdjustSpace{2ex plus 1ex minus -5ex}
          $\begin{array}{l%Equal sign
                       @{\hspace{.1em}}c@{\hspace{.2em}}%Col 3 development
                       l%Equal sign
                       @{\hspace{.1em}}c@{\hspace{.2em}}%Col 5 compact form
                       l%
                      }
         \DevelopSerie{\tfrac{1}{1 - x}}%
                      {1 + x + x^2 + x^3 + x^4 + \cdots}%
                      {\sum_{i=0}^\infty x^i}
         \DevelopSerie{\tfrac{1}{1 - c x}}%
                      {1 + c x + c^2 x^2 + c^3 x^3 + \cdots}%
                      {\sum_{i=0}^\infty c^i x^i}
         \DevelopSerie{\tfrac{1}{1 - x^n}}%
                      {1 + x^n + x^{2n} + x^{3n} + \cdots}%
                      {\sum_{i=0}^\infty x^{ni}}
         \DevelopSerie{\tfrac{x}{(1 - x)^2}}%
                      {x + 2 x^2 + 3 x^3 + 4 x^4 + \cdots}%
                      {\sum_{i=0}^\infty i x^i}
         \DevelopSerie{\sum_{k=0}^n {n \brace k} \tfrac{k! z^k}{(1-z)^{k+1}}}% fix by Raphael Reitzig
                      % x^k {d^n \over dx^n}\left({1 \over 1 - x}\right) % original line
                      {x + 2^n x^2 + 3^n x^3 + 4^n x^4 + \cdots}%
                      {\sum_{i=0}^\infty i^n x^i}
         \DevelopSerie{e^x}%
                      {1 + x + \tfrac{1}{2} x^2 + \tfrac{1}{6} x^3 + \cdots}%
                      {\sum_{i=0}^\infty \tfrac{x^i}{i!}}
         \DevelopSerie{\ln (1 + x)}%
                      {x - \tfrac{1}{2} x^2 + \tfrac{1}{3} x^3 - \tfrac{1}{4} x^4  - \cdots}%
                      {\sum_{i=1}^\infty (-1)^{i+1} \tfrac{x^i}{i}}
         \DevelopSerie{\ln \tfrac{1}{1 - x}}%
                      {x + \tfrac{1}{2} x^2 + \tfrac{1}{3} x^3 + \tfrac{1}{4} x^4  + \cdots}%
                      {\sum_{i=1}^\infty \tfrac{x^i}{i}}
         \DevelopSerie{\sin x}%
                      {x - \tfrac{1}{3!} x^3  + \tfrac{1}{5!} x^5 - \tfrac{1}{7!} x^7 +  \cdots}%
                      {\sum_{i=0}^\infty (-1)^i \tfrac{x^{2i+1}}{(2i+1)!}}
         \DevelopSerie{\cos x}%
                      {1 - \tfrac{1}{2!} x^2  + \tfrac{1}{4!} x^4 - \tfrac{1}{6!} x^6 +  \cdots}%
                      {\sum_{i=0}^\infty (-1)^i \tfrac{x^{2i}}{(2i)!}}
         \DevelopSerie{\tan^{-1} x}%
                      {x - \tfrac{1}{3} x^3  + \tfrac{1}{5} x^5 - \tfrac{1}{7} x^7 +  \cdots}%
                      {\sum_{i=0}^\infty (-1)^i \tfrac{x^{2i+1}}{2i+1}}
         \DevelopSerie{(1+x)^n}%
                      {1 + n x + \tfrac{n(n-1)}{2} x^2 +  \cdots}%
                      {\sum_{i=0}^\infty \binom{n}{i} x^i}
         \DevelopSerie{\tfrac{1}{(1-x)^{n+1}}}%
                      {1 + (n+1) x + \binom{n+2}{2} x^2 +  \cdots}%
                      {\sum_{i=0}^\infty \binom{i+ n}{i} x^i}
         \DevelopSerie{\tfrac{x}{e^x - 1}}%
                      {1 - \tfrac{1}{2} x + \tfrac{1}{12} x^2 - \tfrac{1}{720} x^4 + \cdots}%
                      {\sum_{i=0}^\infty \tfrac{B_i x^i}{i!}}
         \DevelopSerie{\tfrac{1}{2x}(1 - \sqrt{1-4x})}%
                      {1 + x + 2 x^2 + 5 x^3 +  \cdots}%
                      {\sum_{i=0}^\infty \tfrac{1}{i+1}\binom{2i}{i}x^i}
         \DevelopSerie{\tfrac{1}{\sqrt{1-4x}}}%
                      % {1 + x + 2 x^2 + 6 x^3 + \cdots}% % original line
                      {1 + 2 x + 6 x^2 + 20 x^3 + \cdots}% % fix by Raphael Reitzig
                      {\sum_{i=0}^\infty \binom{2i}{i}x^i}
         \DevelopSerie{\tfrac{1}{\sqrt{1-4x}}\left(\tfrac{1 - \sqrt{1-4x}}{2x}\right)^n}%
                      {1 + (2+n)x + \binom{4+n}{2} x^2 + \cdots}%
                      {\sum_{i=0}^\infty \binom{2i+n}{i}x^i}
         \DevelopSerie{\tfrac{1}{1-x}\ln\tfrac{1}{1- x}}%
                      {x + \tfrac{3}{2} x^2 + \tfrac{11}{6} x^3 + \tfrac{25}{12} x^4 + \cdots}%
                      {\sum_{i=1}^\infty H_i x^i}
         \DevelopSerie{\tfrac{1}{2}\left(\ln\tfrac{1}{1- x}\right)^2}%
                      {\tfrac{1}{2} x^2 + \tfrac{3}{4} x^3 + \tfrac{11}{24} x^4 + \cdots}%
                      {\sum_{i=2}^\infty \tfrac{H_{i-1} x^i}{i}}
         \DevelopSerie{\tfrac{x}{1 - x - x^2}}%
                      {x + x^2 + 2 x^3 + 3 x^4 +  \cdots}%
                      {\sum_{i=0}^\infty F_i x^i}
         \DevelopSerie{\tfrac{F_n x}{1 - (F_{n-1} + F_{n+1})x - (-1)^n x^2}}%
                      {F_n x + F_{2n} x^2 + F_{3n} x^3 + \cdots}%
                      {\sum_{i=0}^\infty F_{ni} x^i.}
         \end{array}$
   }
}
