Mathematical equations

This page should have a ton of mathematical equations on it across cells etc.

Inline equations

Here's an inline equation: $\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$. It should be properly rendered within the block of text.

Many equations in once cell

Here's a collection of equations:

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$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$

$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$

They should be rendered properly

Equations across multiple cells w/ mixed content

$$ \sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6} $$

$$\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$

Latex-style equations

\begin{align} \sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\ & = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ & = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\ & = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ & \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right) \end{align}\begin{align} \sqrt{37} & = \sqrt{\frac{73^2-1}{12^2}} \\ & = \sqrt{\frac{73^2}{12^2}\cdot\frac{73^2-1}{73^2}} \\ & = \sqrt{\frac{73^2}{12^2}}\sqrt{\frac{73^2-1}{73^2}} \\ & = \frac{73}{12}\sqrt{1 - \frac{1}{73^2}} \\ & \approx \frac{73}{12}\left(1 - \frac{1}{2\cdot73^2}\right) \end{align}