Convergence of series dealing with factorials

Question

Show the convergence of series :

$$\frac{2\ \cdot\;4\ \cdot\;...\ \cdot\;2n}{3\ \cdot\;5\ \cdot\;...\ \cdot\;(2n+1)}$$

How do I show this series converge? Is there some general technique to follow. I an having a little problem solving these type of questions

Answer 1

Hint: suppose your $n$th term is $f(n)$. Then $f(n) = \dfrac{2n}{2n+1}f(n-1)$. However, $\dfrac{2n}{2n+1} < 1$ always, meaning as $n$ gets bigger $f(n)$ can only get smaller.

Answer 2

Use Raabe's convergence test. It will work.

Answer 3

This series diverges because its general term (Wallis' integral of order $2n+1$) is equivalent to $\frac{\sqrt{\pi}}{2\sqrt{n}}$.