$$\int_{\pi} ^{2 \pi} \frac{|\sin(nx) |} {x} dx\geq\frac{2}{\pi}\sum_{k=1}^n\frac{1}{n+k}$$ I've tried to use basic inequalities such as $$x-\frac{x^{3}}{6} < \sin(x), \quad $$ but it doesn't seem to help. Trying to approximate the right hand sum to ln(2) also didnt help. Please, show me a way to solve it
Asked
Active
Viewed 93 times
2
-
Try to take a look at the Wallis product. – JohannesPauling Mar 02 '21 at 14:18
-
2Same idea as here https://math.stackexchange.com/questions/683576/show-that-int-0-pi-left-frac-sin-nxx-right-mathrmdx-ge-frac2-pi?noredirect=1&lq=1 – Vlad Matei Mar 02 '21 at 14:25
-
There could be another way, but that does seem to be the easy way to do it, if you get the right result (which I suspect you will). – Ian Mar 02 '21 at 14:42
-
Could you present it? I complicated it so much that i cant think of an easy method – Jack Mar 02 '21 at 15:39
-
@Jack Vlad Matei's link works. – River Li Mar 03 '21 at 04:24