<s>[INST] The Formula to this Sequence Series What is the formula for following sequence?
$$\frac12 + \frac12 \cdot \frac34 + \frac12 \cdot \frac34 \cdot \frac56 + ... + \frac12 \cdot \frac34 \cdot \frac56 ... \frac{2n - 1}{2n}$$
This is a question from my Calculus class about sequences.
 [/INST] $$ 
\begin{align} 
&\, 1\,\cdot\,3\,\cdot\,5\,\cdots\,(2n-1) = (2n-1)!! \,,\quad\, 2\,\cdot\,4\,\cdot\,6\,\cdots\,(2n) = (2n)!! \\[2mm] 
&\, S_{\small n} = \frac{1}{2}+\frac{1\,\cdot\,3}{2\,\cdot\,4}+\frac{1\,\cdot\,3\,\cdot\,5}{2\,\cdot\,4\,\cdot\,6}+\,\cdots\,+\frac{1\,\cdot\,3\,\cdot\,5\,\cdots\,(2n-1)}{2\,\cdot\,4\,\cdot\,6\,\cdots\,(2n)} \\[6mm] 
&\, \color{red}{1+S_{\small n}} = 1+\frac{1}{2}+\frac{1\,\cdot\,3}{2\,\cdot\,4}+\frac{1\,\cdot\,3\,\cdot\,5}{2\,\cdot\,4\,\cdot\,6}+\,\cdots\,+\frac{1\,\cdot\,3\,\cdot\,5\,\cdots\,(2n-1)}{2\,\cdot\,4\,\cdot\,6\,\cdots\,(2n)} \\[3mm] 
&\, \qquad = \frac{1\,\cdot\,3}{2}+\frac{1\,\cdot\,3}{2\,\cdot\,4}+\frac{1\,\cdot\,3\,\cdot\,5}{2\,\cdot\,4\,\cdot\,6}+\,\cdots\,+\frac{1\,\cdot\,3\,\cdot\,5\,\cdots\,(2n-1)}{2\,\cdot\,4\,\cdot\,6\,\cdots\,(2n)} \\[3mm] 
&\, \qquad = \frac{1\,\cdot\,3\,\cdot\,5}{2\,\cdot\,4}+\frac{1\,\cdot\,3\,\cdot\,5}{2\,\cdot\,4\,\cdot\,6}+\,\cdots\,+\frac{1\,\cdot\,3\,\cdot\,5\,\cdots\,(2n-1)}{2\,\cdot\,4\,\cdot\,6\,\cdots\,(2n)} \\[3mm] 
&\, \qquad = \,\cdots\, = \frac{1\,\cdot\,3\,\cdot\,5\,\cdots\,(2n+1)}{2\,\cdot\,4\,\cdot\,6\,\cdots\,(2n)} = \color{red}{\frac{(2n+1)!!}{(2n)!!}} \,\Rightarrow\, S_{\small n}=\frac{(2n+1)!!}{(2n)!!}-1 \\[6mm] 
&\, 1+\lim_{n\rightarrow\infty}S_{\small n} = \lim_{n\rightarrow\infty}\frac{(2n+1)!!}{(2n)!!}\color{red}{\,\longrightarrow\,\infty} \qquad \{{\small\text{divergent}}\}
\end{align} 
$$
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