I tried to evaluate
$$ \sum_{k=0}^n a^k\sin(kx) $$
using complex numbers but it didn't work... Any hint?
$a$ and $x$ are real numbers.
2026-04-01 04:42:50.1775018570
How to calculate $\sum_{k=0}^n a^k\sin(kx)$?
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Hint. We assume $a, x\in \mathbb{R}$. Then one may write $$ \sum_{k=0}^n a^k \sin(k x)=\text{Im} \sum_{k=0}^n (ae^{ix})^k =\text{Im}\: \frac{1-(ae^{ix})^{n+1}}{1-ae^{ix}} $$ where we have used the standard evaluation of a geometric sum.