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

# Doubt from trignometry

Look at this https://math.stackexchange.com/questions/13490/proving-that-the-sequence-f-nx-sum-limits-k-1n-frac-sinkxk-is

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

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

@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)