Question: how to calculate $$\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$$
My attempt
Let
$\arctan \theta =\frac{i}{2}\ln \left( \frac{i+\theta }{i-\theta } \right)$
$$S=\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}=\underset{n\to +\infty }{\mathop{\lim }}\,\sum\limits_{k=1}^{n}{\arctan \frac{1}{1+k^{2}}}=\underset{n\to +\infty }{\mathop{\lim }}\,\sum\limits_{k=1}^{n}{\frac{i}{2}\ln \left( \frac{i+\frac{1}{1+k^{2}}}{i-\frac{1}{1+k^{2}}} \right)}$$
$$=\frac{i}{2}\underset{n\to +\infty }{\mathop{\lim }}\,\ln \prod\limits_{k=1}^{n}{\left( \frac{i+\frac{1}{1+k^{2}}}{i-\frac{1}{1+k^{2}}} \right)}=\frac{i}{2}\ln \left( \underset{n\to +\infty }{\mathop{\lim }}\,\prod\limits_{k=1}^{n}{\left( \frac{i+\frac{1}{1+k^{2}}}{i-\frac{1}{1+k^{2}}} \right)} \right)$$
$$=\frac{i}{2}\ln \left( \underset{n\to +\infty }{\mathop{\lim }}\,\prod\limits_{k=1}^{n}{\left( \frac{i\left( 1+k^{2} \right)+1}{i\left( 1+k^{2} \right)-1} \right)} \right)=\frac{i}{2}\ln \left( \underset{n\to +\infty }{\mathop{\lim }}\,\prod\limits_{k=1}^{n}{\left( \frac{1+k^{2}-i}{1+k^{2}+i} \right)} \right)$$
$$=\frac{i}{2}\ln \left( \underset{n\to +\infty }{\mathop{\lim }}\,\prod\limits_{k=1}^{n}{\left( \frac{k^{2}+1-i}{k^{2}+1+i} \right)} \right)=\frac{i}{2}\ln \left( \underset{n\to +\infty }{\mathop{\lim }}\,\prod\limits_{k=0}^{n}{\left( \frac{\left( k+1 \right)^{2}+1-i}{\left( k+1 \right)^{2}+1+i} \right)} \right)$$
Note that $$\left( k+1 \right)^{2}+1\pm i=\left( k+1+i\sqrt{1\pm i} \right)\left( k+1-i\sqrt{1\pm i} \right)$$ $$=\frac{i}{2}\ln \left( \underset{n\to +\infty }{\mathop{\lim }}\,\frac{\prod\limits_{k=0}^{n}{\left( k+1+i\sqrt{1-i} \right)}}{\prod\limits_{k=0}^{n}{\left( k+1+i\sqrt{1+i} \right)}}\cdot \frac{\prod\limits_{k=0}^{n}{\left( k+1-i\sqrt{1-i} \right)}}{\prod\limits_{k=0}^{n}{\left( k+1-i\sqrt{1+i} \right)}} \right)$$