A series problem.

Question

I personally encountered a series problem that I have no idea.

$$
\sum^n_{r=0} \binom{n}{r} \sin[(2r+1)\alpha]
$$

where $\alpha \in (-\frac{1}{2}\pi,\frac{1}{2}\pi)$

My personal guess is that this could probably relate to complex numbers but I don't have a clear idea.

Lunifans · Accepted Answer

$$\begin{aligned}S &= ext{Im}\left[ \sum_{r=0}^{n} \binom{n}{r} e^{i(2r+1)\alpha} ight] \
&= ext{Im}\left[ e^{i\alpha} \sum_{r=0}^{n} \binom{n}{r} (e^{i2\alpha})^r ight]\
&= ext{Im}\left[ e^{i\alpha} (1 + e^{i2\alpha})^n ight]\
&=2^n \cos^n \alpha \sin(n+1)\alpha\
\end{aligned}$$

A series problem.

1 Answers