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finite_sum_of_complex_numbers [2014/01/29 19:55] nikolaj |
finite_sum_of_complex_numbers [2015/06/20 16:22] nikolaj |
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| ===== Finite sum of complex numbers ===== | ===== Finite sum of complex numbers ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#88DDEE: $ (z_i) \in \mathrm{FinSequence}(\mathbb C)$ | | + | | @#55CCEE: context | @#55CCEE: $ (z_i) \in \mathrm{FinSequence}(\mathbb C)$ | |
| - | | @#88DDEE: $ n=\mathrm{length}((z_i)) $ | | + | | @#55CCEE: context | @#55CCEE: $ n=\mathrm{length}((z_i)) $ | |
| + | | @#FFBB00: definiendum | @#FFBB00: $\sum: \mathrm{FinSequence}(\mathbb C)\to \mathbb C$ | | ||
| + | | @#FFBB00: definiendum | @#FFBB00: $\sum_{i=1}^n\ z_i:= \begin{cases} 0 & \mathrm{if}\ n=0\\\\ \left(\sum_{i=1}^{n-1}\ z_i\right)\ +\ z_n & \mathrm{else} \end{cases}$ | | ||
| - | | @#FFBB00: $\sum: \mathrm{FinSequence}(\mathbb C)\to \mathbb C$ | | + | ----- |
| - | | @#FFBB00: $\sum_{i=1}^n\ z_i:= \begin{cases} 0 & \mathrm{if}\ n=0\\\\ \left(\sum_{i=1}^{n-1}\ z_i\right)\ +\ z_n & \mathrm{else} \end{cases}$ | | + | |
| - | + | ||
| - | ==== Discussion ==== | + | |
| === Theorem === | === Theorem === | ||
| - | ^ $\sum_{k=1}^n z^k=\frac{1}{z-1}(z^{n+1}-1)$ ^ | + | ^ $\sum_{k=1}^n z^k=\frac{1}{1-z}(1-z^{n+1})$ ^ |
| - | ==== Parents ==== | + | |
| + | ----- | ||
| === Subset of === | === Subset of === | ||
| [[Finite sum over a monoid]] | [[Finite sum over a monoid]] | ||
