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edited title

edited Jun 29, 2018 at 23:14

Đào Thanh Oai

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$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0$ with $\alpha >1$ and $n=1, 2,\cdots$

Could You give a poof, comment or refrence for the inequality as follows:

$$\sum_{k=1}^n(-1)^k\frac{\sin kx}{k^{\alpha}} < 0$$ for all $n=1,2,3,\ldots$ and $0<x<\pi$ and $\alpha >1$

See also:

$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$

real-analysis inequalities special-functions

asked Jun 29, 2018 at 23:06

Đào Thanh Oai

1.8k
11
37