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$$\sum\limits_{k=1}^n\arctan\frac{ 1 }{ k }=\frac{\pi}{ 2 }$$

Find value of $n$ for which equation is satisfied.

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n=3.

By drawing this figure, you can easily know

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.

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