\documentclass{article}
\begin{document}
$a_{1} x^{2} e^{-\alpha t}
a^{3}_{ij} e^{x^2} \neq {e^x}^2$
x^{n} x_{n}
$\underbrace{ a+b+\cdots+z }_{26}$
\newline
$\Pi$
\newline
$\sqrt{abc}$
\newline
\lim_{x \rightarrow 0} \frac{\sin x}{x}=1
\newline
$$\frac{ x^{2} }{ k+1 }\quad
x^{ \frac{2}{k+1} }\quad x^{ 1/2 }$$
\newline
{n \choose k}
\newline
$$
\sum_{i=1}^{n} \quad
\int_{0}^{\frac{\pi}{2}}
$$
$f(x)=f(a)+\frac{ x-a }{ 1! }f^(^1^)(a)+ \frac{(x-a)^{2}}{2!}f^(^2^)+...+\frac{(x-a)^{n}}{n!}f^(^n^)(a)+Rn(x,a)=\sum_{k=0}^{n} \quad(\frac{(x-a)^{k}}{k!}f^(^k^)(a)+R_n(x,a)$
\newline
$\lim_{x \rightarrow a} \frac{R_n(x,a)}{(x-a)^{n}}=0
\end{document}
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