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+%This command provides the text for the closed form of the sums on the page
+%1 first part of the second column
+\newcommand\TOneSums[1]{%
+ \parbox[t]{#1}{%
+ \TOneSeriesFontSize
+ \begin{DisplayFormulae}{0}{0pt}{\TOneInterlineSeries}{\BigChar}{\StyleWithoutNumber}%
+ \def\FmSep{\unskip\text{,}}
+ \Fm{\sum_{i=1}^n i = \frac{n(n+1)}{2}}
+ \Fm{\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}}
+ \def\FmSep{\relax}
+ \Fm{\sum_{i=1}^n i^3 = \frac{n^2(n+1)^2}{4}}
+ \end{DisplayFormulae}
+
+ \TOneTitle{In general:}
+ \AdjustSpace{-2ex plus .5ex minus .5ex}
+ \begin{DisplayFormulae}{0}{0pt}{\TOneInterlineSeries}{\BigChar}{\StyleWithoutNumber}%
+ %The split of this equation is tricky since it uses a variable length symbol ([)
+ %depending of the size of the sums.
+ %
+ %By using a rule in the first part which has the same depth and height
+ %as the sum symbol it is possible to split the equation
+ %and keep the correct size of the symbol with a variable size.
+ \def\EquationPartB{\sum_{i=1}^n \left((i+1)^{m+1} - i^{m+1} - (m+1)i^m\right)}
+ \settoheight{\TmpLengthA}{$\EquationPartB$}
+ \settoheight{\VSpace}{$\EquationPartB$}
+ \def\FirstPart{\sum_{i=1}^n i^m = \frac{1}{m+1}
+ \left[\mbox{}\rule[\VSpace]{0pt}{\TmpLengthA}\right.}
+ \FmPartA{\FirstPart (n+1)^{m+1} - 1 -}
+ \FmPartB{\FirstPart}{\left.\EquationPartB\right]}
+ \Fm{\sum_{i=1}^{n-1} i^m = \frac{1}{m+1}\sum_{k=0}^m \binom{m+1}{k} B_k n^{m+1-k}}
+ \end{DisplayFormulae}
+
+ \TOneTitle{Geometric series:}
+ \begin{DisplayFormulae}{0}{0pt}{\TOneInterlineSeries}{\BigChar}{\StyleWithoutNumber}%
+ \def\FmSep{\unskip\text{,}}
+ \Fm{\sum_{i=0}^n c^i = \frac{1-c^{n+1}}{1-c}\MathRemark{c \neq 1}}
+ \Fm{\sum_{i=0}^\infty c^i = \frac{1}{1 - c}}
+ \Fm{\sum_{i=1}^\infty c^i = \frac{c}{1 - c}\MathRemark{\vert c \vert < 1}}
+ \Fm{\sum_{i=0}^n i c^i = \frac{nc^{n+2} - (n+1)c^{n+1} + c}{(c-1)^2}\MathRemark{c \neq 1}}
+ \def\FmSep{\relax}
+ \Fm{\sum_{i=0}^\infty i c^i = \frac{c}{(1 - c)^2}\MathRemark{\vert c \vert < 1}}
+ \end{DisplayFormulae}
+
+ \TOneTitle{Harmonic series:}
+ \begin{DisplayFormulae}{0}{0pt}{\TOneInterlineSeries}{\BigChar}{\StyleWithoutNumber}%
+ \def\FmSep{\unskip\text{,}}
+ \Fm{H_n = \sum_{i=1}^n \frac{1}{i}}
+ \Fm{\sum_{i=1}^n iH_i = \frac{n(n+1)}{2}H_n - \frac{n(n-1)}{4}}
+ \Fm{\sum_{i=1}^n H_i = (n+1)H_n - n}
+ \def\FmSep{\relax}
+ \Fm{\sum_{i=1}^n \binom{i}{m} H_i = \binom{n+1}{m+1} \left(H_{n+1} - \frac{1}{m+1}\right)}
+ \end{DisplayFormulae}
+ }%
+}