Problem - 2186
If $\sin t+\cos t=1$, and $s=\cos t +i\sin t$, compute $f(s)=1+s+s^2+\cdots +s^n$
$\sin t + \cos t=1 \implies t=2k\pi$ or $t=2k\pi + \frac{\pi}{2}$, where $k\in\mathbb{Z}$.
When $t=2k\pi$, $s=1$, $f(s)=n+1$.
When $t=2k\pi+\frac{\pi}{2}$, $s=i$, $f(s)=\frac{1+i^{n+1}}{1-i}$
- if $n\equiv 0\pmod{4}$, $f(s)=1$
- if $n\equiv 1\pmod{4}$, $f(s)=1+i$
- if $n\equiv 2\pmod{4}$, $f(s)=i$
- if $n\equiv 3\pmod{4}$, $f(s)=0$