Let $\{a_n\}_{n\ge1}^{\infty}=\bigg\{\cfrac{5\cdot9\cdot13\cdot\dots.\cdot(4n+1)}{7\cdot11\cdot15\cdot\dots.\cdot(4n+3)}\bigg\}$. Find $\lim_{n\to \infty}{a_n}$.
I.) In the first step I studied monotony:
$a_{n+1}-a_{n}=\cfrac{5\cdot9\cdot13\cdot\dots.\cdot(4n+1)}{7\cdot11\cdot15\cdot\dots.\cdot(4n+3)}\cdot\cfrac{-2}{4n+7}<0$, $\{a_n\}$ is decreasing.
II.) In the second step I studied boundary.
$$1>a_{1}=\cfrac{5}{7}>a_{2}=\cfrac{45}{77}>\dots>a_{n}>0$$
III.) In the last step I know that $\{a_n\}$ converges to $a\in\mathbb R$.
$$a_{n+1}=a_{n}\cdot\cfrac{4n+1}{4n+3}$$
Taking the limit as $n\to\infty$:
$$a=a$$
No conclusion.
But if I apply Cesaro-Stolz?
IV.) Let $\{x_n\}_{n\ge1}^{\infty}=\{5\cdot9\cdot13\cdot\dots.\cdot(4n+1)\}$ and $\{y_n\}_{n\ge1}^{\infty}=\{7\cdot11\cdot15\cdot\dots.\cdot(4n+3)\}$. Then
$$\lim_{n\to \infty}{\cfrac{x_{n+1}-x_{n}}{y_{n+1}-y_{n}}=\lim_{n\to \infty}{\cfrac{5\cdot9\cdot13\cdot\dots.\cdot(4n+1)^2}{7\cdot11\cdot15\cdot\dots.\cdot(4n+5)}=?}}$$
If you have a simple solution, I would appreciate it. Thank you!
Best Answer
One can check that $$ \bigg( \frac{4n+1}{4n+3} \bigg)^2 < \frac{n+1}{n+2} $$ for all $n\ge0$. Therefore $$ 0 < a_n = \frac57 \frac9{11} \cdots \frac{4n+1}{4n+3} < \bigg( \frac23 \frac34 \cdots \frac{n+1}{n+2} \bigg)^{1/2} = \sqrt{\frac2{n+2}}, $$ and so $a_n\to0$ by the squeeze theorem.