Let $\displaystyle S_n = \sum_{j=1}^n\sin(\sqrt{j\,}\pi)$. Show $S_{(2M)^2}<0$ and $S_{(2M+1)^2}>0\quad (M=1,2,3,...)$ .

Question

2022-02-08

Let $S_{n} = \sum_{j = 1}^{n} \sin (\sqrt{j} π)$ .

Show $S_{(2 M)^{2}} < 0$ and $S_{(2 M + 1)^{2}} > 0 (M = 1, 2, 3, . . .)$ .

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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;.