How to solve the following summation problem?

Question

$$\sum\limits_{k=1}^n\arctan\frac{ 1 }{ k }=\frac{\pi}{ 2 }$$ Find value of $n$ for which equation is satisfied.

Answer 1

n=3. By drawing this figure, you can easily know

Answer 2

Let use write $$s_n=\sum_{k=1}^n \arctan\frac1k.$$ The sequence $(s_n)_{n\in\mathbf N}$ is increasing. We have $s_0=0$, $s_1=\frac\pi4$ and $s_2=\frac\pi4+\arctan\frac12$. As $\frac12<1$, $\tan^{-1}\left(\frac12\right)<\frac\pi4$ and $s_2<\frac\pi2$. Let us compute $s_3$ using the arctan addition formula $$s_3=\frac\pi4+\arctan\frac12+\arctan\frac13=\frac\pi4+\arctan\frac{\frac12+\frac13}{1-\frac12\frac13}=\frac\pi4+\arctan1=\frac\pi2.$$ $n=3$ is a solution. As $s_4>s_3$, it's the only one.