Positive positive negative negative Series

Question

What is the simplest series that alternates in order $+,+,-,-,+,+,-,- \dots$

Specifically I want to make a Riemann sum for something, but it has this reoccurent pattern I haven't previously encountered. Normally I have seen $(-1)^n$, but this is new to me.

Answer 1

$$f(n) = (-1)^{\dfrac{(n-1)(n + 2)}{2}} = \left\lbrace\begin{matrix}1, & n = 1,2,5,6,\ldots\\-1, & n = 3,4,7,8,\ldots\end{matrix}\right.$$

Answer 2

Depends on what you mean by simplest. The series $\displaystyle\sum_{n=0}^{\infty}a_n$, where $a_n = \operatorname{Re}(i^n) + \operatorname{Im}(i^n)$, follows this pattern.

Answer 3

Hint: the sequence $1,1,-1,-1,1,1,-1,-1,...$ can be described by the formula

$$f(n)=\sin{\frac{\pi n}{2}}+\cos{\frac{\pi n}{2}}.$$