how to get the answer of the following summation

Question

$$\sum\limits_{k=3}^n\arctan\frac{ 1 }{ k }=\frac{\pi}{ 4 }$$ Find value of $n$ for which equation is satisfied.

Answer 1

$$ \sum\limits_{k=3}^5\arctan\frac{ 1 }{ k }\approx \frac{\pi}{ 4 } $$ To prove that, use : $$ \tan(a+b)=\frac{\tan(a)+\tan(b)}{1-\tan(a)\tan(b)} $$

Answer 2

If the summation starts at $k=3$, I have the feeling that there no value of $n$ such that $$f(n)=\sum\limits_{k=3}^n\arctan\frac{ 1 }{ k }-\frac{\pi}{ 4 }$$could be equal to $0$.

$f(n)$ increases with $n$ and you have $$f(3)\simeq -0.463648$$ $$f(4) \simeq -0.218669$$ $$f(5) \simeq -0.021273$$ $$f(6) \simeq +0.143875$$.

If the summation would have started at $k=2$, using Fabien's suggestion, the solution would have been $n=3$.