# Let&nbsp;$$\displaystyle S_n = \sum_{j=1}^n\sin(\sqrt{j\,}\pi)$$. Show&nbsp;$$S_{(2M)^2}&lt;0$$&nbsp;and&nbsp;$$S_{(2M+1)^2}&gt;0\quad (M=1,2,3,...)$$&nbsp;.

2022-02-08

Let ${S}_{n}=\sum _{j=1}^{n}\mathrm{sin}\left(\sqrt{j\phantom{\rule{0.167em}{0ex}}}\pi \right)$.

Show ${S}_{\left(2M{\right)}^{2}}<0$ and ${S}_{\left(2M+1{\right)}^{2}}>0\phantom{\rule{1em}{0ex}}\left(M=1,2,3,...\right)$ .

You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it