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Diffstat (limited to 'math/math.tex')
| -rw-r--r-- | math/math.tex | 62 |
1 files changed, 31 insertions, 31 deletions
diff --git a/math/math.tex b/math/math.tex index 430d533..cc8d263 100644 --- a/math/math.tex +++ b/math/math.tex @@ -295,6 +295,37 @@ Anzahl der Teilmengen von $\mathbb{N}$, die sich zu $n$ aufaddieren mit maximale \end{tabular} \vspace{5mm} +\begin{tabular}{l|l|l} + \toprule + \multicolumn{3}{c}{Reihen} \\ + \midrule + $\sum\limits_{i = 1}^n i = \frac{n(n+1)}{2}$ & + $\sum\limits_{i = 1}^n i^2 = \frac{n(n + 1)(n + 2)}{6}$ & + $\sum\limits_{i = 1}^n i^3 = \frac{n^2 (n + 1)^2}{4}$ \\ + + $\sum\limits_{i = 0}^n c^i = \frac{c^{n + 1} - 1}{c - 1} \quad c \neq 1$ & + $\sum\limits_{i = 0}^\infty c^i = \frac{1}{1 - c} \quad \vert c \vert < 1$ & + $\sum\limits_{i = 1}^\infty c^i = \frac{c}{1 - c} \quad \vert c \vert < 1$ \\ + + \multicolumn{2}{l|}{ + $\sum\limits_{i = 0}^n ic^i = \frac{nc^{n + 2} - (n + 1)c^{n + 1} + c}{(c - 1)^2} \quad c \neq 1$ + } & + $\sum\limits_{i = 0}^\infty ic^i = \frac{c}{(1 - c)^2} \quad \vert c \vert < 1$ \\ + + $H_n = \sum\limits_{i = 1}^n \frac{1}{i}$ & + \multicolumn{2}{l}{ + $\sum\limits_{i = 1}^n iH_i = \frac{n(n + 1)}{2}H_n - \frac{n(n - 1)}{4}$ + } \\ + + $\sum\limits_{i = 1}^n H_i = (n + 1)H_n - n$ & + \multicolumn{2}{l}{ + $\sum\limits_{i = 1}^n \binom{i}{m}H_i = + \binom{n + 1}{m + 1} \left(H_{n + 1} - \frac{1}{m + 1}\right)$ + } \\ + \bottomrule +\end{tabular} +\vspace{5mm} + \begin{tabular}{c|cccc} \toprule \multicolumn{5}{c}{The Twelvefold Way (verteile $n$ Bälle auf $k$ Boxen)} \\ @@ -505,37 +536,6 @@ Anzahl der Teilmengen von $\mathbb{N}$, die sich zu $n$ aufaddieren mit maximale \end{tabular} \vspace{5mm} -\begin{tabular}{l|l|l} - \toprule - \multicolumn{3}{c}{Reihen} \\ - \midrule - $\sum\limits_{i = 1}^n i = \frac{n(n+1)}{2}$ & - $\sum\limits_{i = 1}^n i^2 = \frac{n(n + 1)(n + 2)}{6}$ & - $\sum\limits_{i = 1}^n i^3 = \frac{n^2 (n + 1)^2}{4}$ \\ - - $\sum\limits_{i = 0}^n c^i = \frac{c^{n + 1} - 1}{c - 1} \quad c \neq 1$ & - $\sum\limits_{i = 0}^\infty c^i = \frac{1}{1 - c} \quad \vert c \vert < 1$ & - $\sum\limits_{i = 1}^\infty c^i = \frac{c}{1 - c} \quad \vert c \vert < 1$ \\ - - \multicolumn{2}{l|}{ - $\sum\limits_{i = 0}^n ic^i = \frac{nc^{n + 2} - (n + 1)c^{n + 1} + c}{(c - 1)^2} \quad c \neq 1$ - } & - $\sum\limits_{i = 0}^\infty ic^i = \frac{c}{(1 - c)^2} \quad \vert c \vert < 1$ \\ - - $H_n = \sum\limits_{i = 1}^n \frac{1}{i}$ & - \multicolumn{2}{l}{ - $\sum\limits_{i = 1}^n iH_i = \frac{n(n + 1)}{2}H_n - \frac{n(n - 1)}{4}$ - } \\ - - $\sum\limits_{i = 1}^n H_i = (n + 1)H_n - n$ & - \multicolumn{2}{l}{ - $\sum\limits_{i = 1}^n \binom{i}{m}H_i = - \binom{n + 1}{m + 1} \left(H_{n + 1} - \frac{1}{m + 1}\right)$ - } \\ - \bottomrule -\end{tabular} -\vspace{5mm} - \begin{tabular}{ll} \toprule \multicolumn{2}{c}{Verschiedenes} \\ |