Actually basic rules of TeX notation are very simple.
To insert formula into the text it should be surrounded with $ signs:
text.. 
..text
If you want to obtain a formula on a separate string then it should be surrounded with $ signs:
text..
..text
Keywords begin with \. Functional keywords:
\int for integral
\sum for summation
\frac for fraction, etc (\sin, \log, \prod...)
Keywords for math. signs:
A \subset B - A is a subset of B
x \in A - x is an element of A
A \cap B - intersection of A and B
\delta, \phi, \lambda - Greek literals
etc., there are 100-200 signs
Sub- and superscripts:
keyword_subscript^superscript. For example
x^3 - x cubed,
\int_a^b - integral from a to b
Braces. Parts of formula, which are formulas themselves, should be surrounded with braces:
\sum_{i=1}^n - sum over i from 1 to n
![Дано
$$
I_{n,1}=O(|f'(a_1)-n|^{-1}+|f'(a)-n|^{-1}),
I_{n,2}=O(|f'(b_1)-n|^{-1}+|f'(b)-n|^{-1}),
$$
Доказать, что
$$
\sum_{f'(a)-\Delta\leq n\leq f'(b)+\Delta}(I_{n,1}+I_{n,2})=O(\log(f'(b)-f'(a)+2))
$$
где $a_1=[a]+1.5,\ b_1=[b]-0.5$](https://dxdy-01.korotkov.co.uk/f/4/e/3/4e3c4cd83e30662c7f0d0a367a2a0abd82.png "Дано
$$
I_{n,1}=O(|f'(a_1)-n|^{-1}+|f'(a)-n|^{-1}),
I_{n,2}=O(|f'(b_1)-n|^{-1}+|f'(b)-n|^{-1}),
$$
Доказать, что
$$
\sum_{f'(a)-\Delta\leq n\leq f'(b)+\Delta}(I_{n,1}+I_{n,2})=O(\log(f'(b)-f'(a)+2))
$$
где $a_1=[a]+1.5,\ b_1=[b]-0.5$")
[/math]
for advanced and professional typesetting.
Any TeX-based tysetting system claims attentiveness and persistence within creating documents from its users, because nothing is predefined! I guess, in the Inet there is plenty of appropriate guide books devoted to TeX, LaTeX, teTeX, OzTeX etc. (depending on the OS), some of them being contained at this outstanding elibrary.