\documentclass[12pt]{article}

% COPY/PASTE THIS PREAMBLE INTO OVERLEAF
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amscd,mathrsfs,amscd,amsthm}
\usepackage[pdftex]{graphicx,xcolor}
\usepackage[nohead,margin=1in]{geometry} % Page geometry




% Your LaTeX code must be written between \begin[document} ...... \end{document}

\begin{document}

\textbf{Setup}

\begin{enumerate}
\item Create an account on \texttt{www.overleaf.com}
\item Choose a template.
\item Copy paste the Preamble in this TeX document into the left hand screen of \texttt{overleaf}
\item Ask me if you have questions.
\item Use Google if you want to find out how to do X, Y, Z (someone has already asked your question, believe me!)
\end{enumerate}

\textbf{\LaTeX commands}

To input basic text you need to type.

You can make text \textbf{bold}, \textit{italicized} (v.1), \textsl{italicised} (v.2) or \textsc{small caps}. You can also type in \texttt{typewriter type}.

You can make text {\large large}, {\Large larger}, {\LARGE largest}, {\huge largestest}, {\Huge largestestest}, {\small small}, {\footnotesize smaller} or {\tiny tiny}.

\noindent
You can take away an indent \hspace{3cm} add horizontal space

\vspace{4cm}
or vertical space.
\begin{center}
Text can also {\color{blue} be} {\color{red} colourful} and centered (but you need to ensure you include the appropriate package).
\end{center}

\medskip
\noindent
\textbf{Mathematics:}

You can include sequences $(a_{n})$ in text, or as displayed math
	\[(a_{n})\]
You can also write limits $\lim_{n\to \infty} a_{n}$ in text, and general symbols $\Sigma$, $\infty$, $\alpha$, $\pi$, $\Pi$.

Fractions are $\frac{easy}{to}$ write $\frac{2}{3}$ or $\frac{\pi}{2}$, or	
	\[ \frac{2345}{3456}, \quad \frac{2\cdot 3\cdot \cdots \cdot 5}{3.4\cdot7 \cdot \cdots 8}\]

And we can write nice integrals $\int_{0}^{1} f(x)dx $ in text, or displayed
	\[ \int_{a}^{b}\int_{c}^{d} \sqrt{4-x^{2}} dx = \frac{d}{dz} \int  \sqrt{4+x^{2}}\frac{dx}{dt}\]


\textbf{Example Template}

Let $(b_{n})$ be a sequence of nonzero real numbers.

\begin{enumerate}
\item The $m^{th}$ \textbf{partial product associated to $(b_{n})$} is 
	\[ p_{m} = b_{1}b_{2}\cdots b_{m}.\]
\item The \textbf{sequence of partial products associated to $(b_{n})$} is the sequence $(p_{m})$, where $p_{m}$ is the $m^{th}$ partial product associated to $(b_{n})$.
\item If the sequence $(p_{m})$ of partial products associated to $(b_{n})$ is convergent and $L=\lim_{m\to\infty} p_{m} \neq 0$, then we say that
	\[ \prod_{n=1}^{\infty} b_{n} = L = \lim_{m\to \infty} p_{m}.\]
We call $\prod_{n=1}^{\infty} b_{n}$ an \textbf{infinite product}. In this case, we say the infinite product $\prod_{n=1}^{\infty} b_{n}$ \textbf{converges}; otherwise, the infinite product \textbf{diverges}. In particular, if $\lim p_{m} =0$ then the infinite product diverges.
\end{enumerate}

\medskip
\noindent
\textbf{Example:}
\begin{enumerate}
\item Let $b_{n}=\frac{n}{n+1} = 1+\frac{1}{n}$. Then,	
	\[ b_{1}=\frac{1}{2},\;b_{2}=\frac{2}{3},\; b_{3}=\frac{3}{4},\ldots\]
The partial products associated to $(b_{n})$ are
	\[ p_{1} = b_{1} = \frac{1}{2},\; p_{2}=b_{1}b_{2}=\frac{1}{3},\; p_{3} =b_{1}b_{2}b_{3} = \frac{1}{4},\ldots \]
In general,
	\[ p_{m} =b_{1}b_{2}\cdots b_{m} = \frac{1}{2}\cdot \frac{2}{3}\cdot \frac{3}{4}\cdots \frac{m-1}{m}\cdot \frac{m}{m+1} = \frac{1}{m+1}\]
Hence, $\lim_{m\to \infty} p_{m} =0$ and the infinite product diverges.
\end{enumerate}
\end{document}