% LESSON SEVEN
          
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	\textbf{Lesson Seven: Math Miscellany}
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A consequence of the Pythagorean Theorem is the fact that 
$\sin^2\theta + \cos^2\theta = 1$.\\
% Compare the appearance if you use sin instead of \sin.
% LaTeX has built-in versions of all the common math 
%   functions: \ln, \arctan, etc.

One of the most amazing integrals is the following. 
Who'd have thought that $\pi$ would pop in there?!?

$$\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}$$
% The \, inserts a 'thinspace.'
% \: gets a 'medspace' and \; produces a 'thickspace.'
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% a text containing a comma, colon, or semicolon. Brilliant, no?

% To produce a cube root, use \sqrt[3]{x}. 
% I leave it to you to guess how to get a fourth root.

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...

$$\sum_{n=1}^{\infty}\frac{1}{n^2}
=\frac{1}{1}+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\dots=\frac{\pi^2}{6}$$
% If you just type ... instead of \dots, the placement is off.

Some of our favorite numbers are 
$\displaystyle e=\lim_{n \rightarrow \infty}(1+1/n)^n$ 
and $\displaystyle\ln 2 = \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}$.\\
% Compare with the appearance when \displaystyle is omitted.

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