<s>[INST] Series question,related to telescopic series, 1/2*4+ 1*3/2*4*6+ 1*3*5/2*4*6*8 ...infinity The series is $$\frac{1}{2*4}+ \frac{1*3}{2*4*6}+ \frac{1*3*5}{2*4*6*8}+....$$ 
It continues to infinity.I tried multiplying  with $2$ and dividing each term by$(3-1)$,$(5-3)$ etc,starting from the second term which gives me $$\frac{1}{8} -\frac{1}{4*6} -\frac{1}{4*6*8}-\frac{1}{4*6*8*10}$$.
Also if anybody is wondering the answer 0.5
 [/INST] We have
$$
\begin{align}
\frac{1\cdot 3\cdot 5 \cdots(2n-1)}{2\cdot 4\cdot 6\cdots(2n+2)}&=\frac{1\cdot 2\cdot 3\cdot 4 \cdot 5\cdot6\cdots(2n-1)\cdot 2n}{(2\cdot 4\cdot 6\cdots (2n))^2(2n+2)}\\\\
&=\frac{(2n)!}{(2^{n} \cdot 1\cdot 2\cdot 3 \cdot 4 \cdots n)^2 \cdot (2n+2)}\\\\
& =\frac{(2n)!}{2^{2n} (n!)^2 \cdot (2n+2)}\\\\
&=\frac{1}{(2n+2)\,4^n}\binom{2n}{n}\\\\&=\frac{1}{4^n}\binom{2n}{n}-\frac{1}{4^{n+1}}\binom{2n+2}{n+1}.
\end{align}
$$
The series is then telescopic:
$$\sum_{n=1}^{N}\frac{1\cdot 3\cdot 5 \cdots(2n-1)}{2\cdot 4\cdot 6\cdots(2n+2)}=\frac{1}{2}-\frac{1}{4^{N+1}}\binom{2N+2}{N+1}.
$$As $N$ is great, one may prove that
$$\frac{1}{4^{N+1}}\binom{2N+2}{N+1}=O\left(\frac{1}{\sqrt{N}}\right).
$$ 
The series is thus equal to $\dfrac{1}{2}$:

$$\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5 \cdots(2n-1)}{2\cdot 4\cdot 6\cdots(2n+2)}=\frac{1}{2}.
$$

</s>