diff options
Diffstat (limited to 'info/mathtrip/src/sums.tex')
-rw-r--r-- | info/mathtrip/src/sums.tex | 54 |
1 files changed, 54 insertions, 0 deletions
diff --git a/info/mathtrip/src/sums.tex b/info/mathtrip/src/sums.tex new file mode 100644 index 0000000000..9e4a01cd04 --- /dev/null +++ b/info/mathtrip/src/sums.tex @@ -0,0 +1,54 @@ +%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} + }% +} |