We've been given the sequence $x_n=(-1)^n \cdot \frac{\sqrt{n}}{n+1}+\sin \frac{n \pi}{2}$.
I have to find the min, max, inf and sup (if they exist), and also find the points of accumulation.
I have no idea how to do that because I only worked with simple sequences that are monotone increasing or decreasing, and this one isn't so easy.
$$\mathrm{{d\over dx}{x^{0.5}\over(1+x)}=-{x^{0.5}\over(1+x)^2}+{0.5\over x^{0.5}(1+x)}={0.5(1-x)\over x^{0.5}(1+x)^2}<0\;\forall\;x>1}$$ So $\mathrm{\sqrt{n}\over n+1}$ is strictly decreasing.$\quad\cdots(1)$
Also $$\mathrm{\lim_{n\to\infty}{\sqrt{n}\over n+1}=0}\quad\cdots(2)$$ $\mathrm{\sin\left({k\pi\over2}\right)=0,1,0,-1}$ at $\mathrm{k=4n,4n+1,4n+2,4n+3}$ respectively.$\quad\cdots(3)$
Using $(1),(2),(3)$ we see $$ \mathrm{\sup_{n\ge0}x_{4n}={2\over5}},\quad\mathrm{\sup_{n\ge0}x_{4n+2}={\sqrt{2}\over3}},\quad \mathrm{\sup_{n\ge0}x_{4n+1}=1},\quad\mathrm{\sup_{n\ge0}x_{4n+3}=-1}\\ \mathrm{\inf_{n\ge0}x_{4n}=0},\quad\mathrm{\inf_{n\ge0}x_{4n+2}=0},\quad \mathrm{\inf_{n\ge0}x_{4n+1}={1\over2}},\quad\mathrm{\inf_{n\ge0}x_{4n+3}=-{\sqrt{3}\over4}-1} $$ So $$\mathrm{\sup_{n\ge0}x_n=1,\quad\inf_{n\ge0}x_n=-{\sqrt{3}\over4}-1}$$ and using $(2),(3)$ the accumulation points are $0,\pm1$.