----- > [!proposition] Proposition. ([[finite geometric series]]) > Let $w \in \mathbb{C}$, $w \neq 1$. Then $\sum_{n=0}^{N} w ^{n} = \frac{1-w^{N+1}}{1-w}$ and if $w \neq 0$ $\sum_{n=-N}^{N} w ^{n} = \frac{w ^{-N} - w ^{N+1}}{1-w}.$ It follows $\sum_{n=1}^{N} w^{n}=\frac{w(1-w^{N})}{1-w}.$ > (If $w=1$ then the summation is immediate to compute.) > [!proof]- Proof. ([[finite geometric series]]) > 1. Consider $\begin{align} (1-w)\left( \sum_{n=0}^{N} w ^{n} \right ) = & \sum_{n=0}^{N} w ^{n}- \sum_{n=0}^{N} w ^{n+1} \\ > = & 1 \cancel{- w + w - \dots + w ^{N}}^{\text{telescoping}} - w ^{N+1}. > \end{align}$ > Then divide both sides by $(1-w)$. > > 2. Consider $\begin{align} > (1-w)\sum_{n=-N}^{N} w ^{n} = & \sum_{n=-N}^{N} w ^{n} - \sum_{n=-N}^{N} w ^{n+1} \\ > = & w ^{-N} \cancel{- w ^{-N+1} + w ^{-N+1} - \dots + w ^{N}} ^{\text{telescoping}} - w ^{N+1} . > \end{align}$ > > To get the identity starting at $n=1$, just subtract of $w^{0}=1$ from the formula (1) and simplify. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```