Doubt from trignometry

So we can now conclude that if x \epsilon [0,π] \displaystyle \left|{\sum_{k=1}^n\frac{sin(kx)}{k}}\right| \approx \frac{π-x}{2}<2\sqrt{π}
For high n

Look at this

How maximum value is {2\sqrt{\pi}} ?? Why u take n high value even question tell any positive value

Bro if x\epsilon R then my greatest integer function will add up
I already posted it

I can insure you that function will never be equal to 2\sqrt{π} @Sneha_2021

Yeah we can conclude that fact but still can't prove it... i thought of pursuing using beta gamma function but did not yeild much results

Bro also Upper bound that function will be very close to y=\frac{π}{2} As n\rightarrow \infty
X value for any quadrant

@Samrat_2020 @Sneha_2021 Look at this

@Sneha_2021 this is not in course( Abel's inequality require here and some lemmas) i mean it requires high knowledge.
Even if u want i can share solution
Reply fast. Otherwise not able to join after some time ( for some reasons)

Ok. No problem. I am sharing


Thanku @Tushar_2019 broo now get some depth idea nd sorry for late reply not active in site