1\documentclass{article}
2\begin{document}
3
4This is an equation environment in which to test fractions.  Not great translation
5but at least some.   This uses \verb#\frac#
6$$
7\frac{1}{2} = \frac{\alpha+\beta}{\gamma+\delta}
8$$
9This is an equation environment in which to test fractions.  Not great translation
10but at least some.  This uses \verb#\over#
11$$
12{1 \over 2} = {\alpha+\beta \over \gamma+\delta}
13$$
14Now I will test the square root.  First a simple square root
15$$
16z = \sqrt{x+y}
17$$
18now something a bit harder
19$$
20z = \sqrt{x^2+y^2}
21$$
22and what the heck, lets put a square root together with a fraction
23$$
24z = \frac{1}{\sqrt{x+y}}
25$$
26Fractions of the form
27\[ \frac{\mbox{\textit{numerator}}}{\mbox{\textit{denominator}}} \]
28are obtained in \LaTeX\ using the construction (This was once a bug.)
29
30Now for some integrals.  How about a somewhat complicated one to start
31$$
32\int_{0}^{\infty} \frac{a}{\sqrt{x^2+c}}\,dx
33$$
34and
35$$
36\int_0^\infty \frac{a}{\sqrt{x^2+c}}\,dx
37$$
38and now using \verb#\limits#
39$$
40\int \limits_0^\infty \frac{a}{\sqrt{x^2+c}}\,dx
41$$
42A summation example
43$$
44\sum_{n=0}^\infty \frac{a_n}{\sqrt{x_n^2+c_n}}\,dx
45$$
46A product example
47$$
48\sum_{n=0}^\infty \frac{a_n}{\sqrt{x_n^2+c_n}}\,dx
49$$
50\end{document}
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