Is $\sin^2(p\pi+\frac{\pi}{2})$ a Cauchy sequence?

Question

The question is $(C_p)_{p\in\mathbb{N}}, sin^2(p\pi+\frac{\pi}{2})$ a Cauchy sequence? I've worked out the first few terms, they are $6.75\times10^{-3}, 0.0187, 0.0364, 0.0594, 0.0882, 0.122, 0.19, ...$. I'm sure that the terms will decrease later and then increase again (according to the graph of sine). This makes me feel like this is not a Cauchy sequence. Am I correct? Is this the way I should use to determine if a sequence is Cauchy or not?

Answer

I'm sorry y'all this was so silly... I used degrees instead of radian. The sequence should be $(1, 1, 1, ...)$.

Answer 1

$\sin(p \pi+\frac{\pi}{2})=\cos(p \pi)$. Then, your sequence is the constant sequence $(1, 1, 1, \ldots)$ wich is convergent and consequently, is a Cauchy sequence.

Answer 2

$\sin^2{(p\pi+\frac{\pi}{2})}=\frac{1}{2}(1-\cos{(2p\pi+\pi)})=\frac{1}{2}(1-\cos{((2p+1)\pi)$

for which if $p$ is a natural number is always $1$.

so it is a constant sequence for all $p$.

therefore it is cauchy.