$$ \arctan\left(1 \over 3\right) + \arctan\left(1 \over 7\right) + \arctan\left(1 \over 13\right) + \arctan\left(1 \over 21\right) + \cdots $$ Estimate the value of the expression if $n \to \infty$, how to derive this and what will be the approach of this kind of question ?
I have tried to solve this by first doing partial sums then taking limit but I can't evaluate this after getting the form $\left(n^{2} + n + 1\right)$.
$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} &\lim_{N \to \infty}\,\,\,\sum_{k = 0}^{N}\arctan\pars{1 \over k^{2} + 3k + 3} = \lim_{N \to \infty}\,\,\,\sum_{k = 0}^{N} \bracks{\arctan\pars{k + 2} - \arctan\pars{k + 1}} \\[1cm] = &\ \lim_{N \to \infty}\left\lbrace\vphantom{\Large A}% \bracks{\vphantom{\large A}\arctan\pars{2} - \arctan\pars{1}} + \bracks{\vphantom{\large A}\arctan\pars{3} - \arctan\pars{2}} + \cdots\right. \\[5mm] &\ \left.\phantom{\lim_{N \to \infty}}% \mbox{} + \bracks{\vphantom{\large A}\arctan\pars{N + 2} - \arctan\pars{N + 1}}\right\rbrace \\[8mm] = &\ \lim_{N \to \infty}\,\,\, \bracks{\vphantom{\large A}-\arctan\pars{1} + \arctan\pars{N + 2}} = -\,{\pi \over 4} + {\pi \over 2} = \bbx{\ds{\pi \over 4}} \end{align}