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\textbf{Lesson Seven: Math Miscellany}
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A consequence of the Pythagorean Theorem is the fact that
$\sin^2\theta + \cos^2\theta = 1$.\\
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One of the most amazing integrals is the following.
Who'd have thought that $\pi$ would pop in there?!?
$$\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}$$
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And can you believe what happens when you sum the reciprocals
of all the perfect squares? What on earth could that have to do
with circles? And yet $\pi$ makes another appearance...
$$\sum_{n=1}^{\infty}\frac{1}{n^2}
=\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\dots=\frac{\pi^2}{6}$$
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Some of our favorite numbers are
$\displaystyle e=\lim_{n \rightarrow \infty}(1+1/n)^n$
and $\displaystyle\ln 2 = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}$.\\
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