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title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><p>In this example, we compute rigorous bounds on <span>$\pi$</span> using interval arithmetic<sup class="footnote-reference"><a id="citeref-1" href="#footnote-1">[1]</a></sup>, via the <a href="https://github.com/JuliaIntervals/IntervalArithmetic.jl"><code>IntervalArithmetic.jl</code></a> package.</p><p>There are many ways to calculate <span>$\pi$</span>. For illustrative purposes, we will use the following sum</p><p class="math-container">\[S \bydef \sum_{n=1}^\infty \frac{1}{n^2}.\]</p><p>According to the <a href="https://en.wikipedia.org/wiki/Basel_problem">Basel Problem</a>, the exact value is <span>$S = \frac{\pi^2}{6}$</span>. Thus, if we can calculate a rigorous enclosure of <span>$S$</span>, then we can deduce a rigorous enclosure of <span>$\pi$</span>.</p><p>First, we split <span>$S$</span> into a finite and infinite part, <span>$S = S_N + T_N$</span>, where</p><p class="math-container">\[S_N \bydef \sum_{n=1}^N \frac{1}{n^2}, \qquad
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