Closed form of a sequence containing the pattern $\{+,-,-,+,+,-,-,...\}$

Question

I was in the middle of answering this question that asked how to get the Taylor series of $cos(x)$ centered at $a=\frac{\pi}{3}$. I was reaching the point where I was about to write $(-1)^n$ making the it an alternating series, when I began to realize why calculus II courses only talk about the Maclaurin series of $cos(x)$. In the Maclaurin series when $a=0$, the function alternates signs every iteration using $(-1)^n$, and has the pattern of $\{-,+,-,+,-,+\}$. However, when the function is centered at $a=\frac{\pi}{3}$, the series has the pattern $\{+,-,-,+,+,-,-,+,+,...\}$
What form of $(-1)^{kn}$ can create a sequence with this kind of pattern?

Answer 1

For sequences of period 4, one has to rely on the fourth roots of unity, that is, to linear combinations of the sequences of general term $1$, $\mathrm i^n$, $(-1)^n$ and $(-\mathrm i)^n$. In your case, starting at $n=1$, the sequence is $$ \frac{1-\mathrm i}2\cdot\mathrm i^n+\frac{1+\mathrm i}2\cdot(-\mathrm i)^n. $$ Another way to express the same sequence is to use cosine and sine functions with period 4, in your case, still starting at $n=1$, one would consider $$ \sqrt2\cdot\cos\left(n\frac\pi2-\frac\pi4\right). $$

Answer 2

$$(-1)^{\lceil n/2\rceil}$$ where $\lceil x\rceil$ is the ceiling function, i.e. the smallest integer $\ge x$.