No mathematical paper can be written without use of at least one of the following symbols (expresions):
Superscripted or/and subscripted symbols are typeset with the caret ^ and the underscore _.
Superscripts are very useful when you talk for example about the Pythagoras formula $a^2+b^2=c^2$, while subscripts can be helpful in talking about terms of sequences or indices: $x_{k}$, $a_{k_{p}}$ and $j_i$.
The \frac{}{} command takes two arguments: the numerator and denominator and typesets them in normal fraction form:
To multiply $\frac{3}{4}$ by $\frac{1}{2}$ we multiply the two numerators and the two denominators, respectively, and we obtain $\frac{3}{8}$. In other words: \[ \frac{3}{4} \cdot \frac{1}{2} = \frac{3 \cdot 1}{4 \cdot 2}=\frac{3}{8}. \]
The square root is entered as \sqrt{} ; the nth root is generated with \sqrt[n]{} :
For most numbers, the square root is irrational, e.g. \[ \sqrt{2}=1.414212563\ldots \] However, it is quite easy to see that $\sqrt[3]{8}=2$ and $\sqrt[5]{-1}=-1$.
The sum, product and integral operator are produced by the commands \sum_{}^{} , \prod_{}^{} and \int_{}^{} ,
where the lower and upper limits are specified with _ and ^.
A geometric series it the one where each successive term is produced by multiplying the previous term by a constant \[ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8} + \frac{1}{16}+\ldots = \sum_{n=0}^{\infty}\frac{1}{2^n}. \] Form Tychonoff's theorem we know that if each topological space $X_{\lambda}$, where $\lambda \in \Lambda$, is compact then the product \[ X=\prod_{\lambda \in \Lambda} X_{\lambda} \] is compact. If $f$ is a continuous real-valued function defined on a closed interval $[a,b]$, and $F$ is its antiderivative, the definite integral of $f$ over that interval is given by \[ \int_{a}^{b} f(x) \textup dx = F(b)-F(a). \]
If you would like to use multiple integrals, just say \iint, \iiint, \iiiint, or \idotsint.
\[ \iint_{\Omega} (2x^2-y^3+xy)\textup dx \textup dy = \iint_{\Omega} 2x^2\textup dx \textup dy - \iint_{\Omega} y^3\textup dx \textup dy + \iint_{\Omega} xy\textup dx \textup dy. \] and \[ A=\idotsint_{\Omega} f(x_1,x_2,\ldots,x_n) \]
When writing in TeX you can use parentheses ( and ), brackets [ and ], braces \{ and \} and many other delimiters:
Schwarz proved that for any $x,y \in X$ the following inequality holds \[ |\langle x,y \rangle| \leq \|x\| \|y\|. \] In some cases it is necessary to specify the size of delimiters. This can be done with the commands: {{{\bigl}}}, {{{\Bigl}}}, {{{\biggl}}}, {{{\Biggl}}} and simirarly {{{\bigr}}}, {{{\Bigr}}}, {{{\biggr}}}, {{{\Biggr}}}. In case of real-valued square integrable functions Schwarz inequality says that \[ \int_{0}^{1} f(x)g(x) \textup dx \leq \biggl( \int_{0}^{1} \bigl| f(x) \bigr|^2 \textup dx \biggr)^{\frac{1}{2}} \biggl( \int_{0}^{1} \bigl| g(x) \bigr|^2 \textup dx \biggr)^{\frac{1}{2}}. \]
Instead of \bigl, \Bigl, … you can use the \left and \right commands, which are put in front of the delimeter. These commands will automatically determine the correct size of the delimiter, however in that case the delimiters very often turns out larger than necessary, for example:
\[ \left[ \sum_{i} a_{i} \left| \sum_{j} x_{ij} \right|^p \right]^{1/p} \qquad \text{versus} \qquad \biggl[ \sum_{i} a_{i} \Bigl| \sum_{j} x_{ij} \Bigr|^p \biggr]^{1/p} \]