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diff --git a/math/tables/series.tex b/math/tables/series.tex new file mode 100644 index 0000000..13af68f --- /dev/null +++ b/math/tables/series.tex @@ -0,0 +1,33 @@ +\begin{tabularx}{\linewidth}{|XIXIXIX|} + \hline + \multicolumn{4}{|c|}{Reihen} \\ + \hline + $\sum\limits_{i = 1}^n i = \frac{n(n+1)}{2}$ & + $\sum\limits_{i = 1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}$ & + $\sum\limits_{i = 1}^n i^3 = \frac{n^2 (n + 1)^2}{4}$ & + $H_n = \sum\limits_{i = 1}^n \frac{1}{i}$ \\ + \grayhline + + $\sum\limits_{i = 0}^n c^i = \frac{c^{n + 1} - 1}{c - 1} \quad c \neq 1$ & + $\sum\limits_{i = 0}^\infty c^i = \frac{1}{1 - c} \quad \vert c \vert < 1$ & + $\sum\limits_{i = 1}^\infty c^i = \frac{c}{1 - c} \quad \vert c \vert < 1$ & + $\sum\limits_{i = 0}^\infty ic^i = \frac{c}{(1 - c)^2} \quad \vert c \vert < 1$ \\ + \grayhline + + \multicolumn{2}{|lI}{ + $\sum\limits_{i = 0}^n ic^i = \frac{nc^{n + 2} - (n + 1)c^{n + 1} + c}{(c - 1)^2} \quad c \neq 1$ + } & + \multicolumn{2}{l|}{ + $\sum\limits_{i = 1}^n iH_i = \frac{n(n + 1)}{2}H_n - \frac{n(n - 1)}{4}$ + } \\ + \grayhline + + \multicolumn{2}{|lI}{ + $\sum\limits_{i = 1}^n H_i = (n + 1)H_n - n$ + } & + \multicolumn{2}{l|}{ + $\sum\limits_{i = 1}^n \binom{i}{m}H_i = + \binom{n + 1}{m + 1} \left(H_{n + 1} - \frac{1}{m + 1}\right)$ + } \\ + \hline +\end{tabularx}
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