Limit of a Sequence

• June 30th 2010, 12:26 PM
Limit of a Sequence
I'm having trouble with the sequence: $a_n=\frac{1*3*5*...*(2n-1)}{n!}$

It seems that the limit must exists since each factor $a_n=\frac{1}{1}*\frac{3}{2}*\frac{5}{3}*\frac{2n-1}{n}$ has a finite limit.
• June 30th 2010, 12:41 PM
HallsofIvy
Write 1*3*5*...(2n-1) as $\frac{1*2*3*4*5*...*(2n-1)(2n)}{2*4*6*...**(2n)}= \frac{(2n)!}{((1)(2))(2(2))*(3(2))*...*(n(2))$ $=\frac{(2n)!}{2^n n!}$.

Now, $a_n= \frac{(2n)!}{2^n(n!)^2}$.
• June 30th 2010, 01:05 PM
Thanks, but you have a latex error.
• June 30th 2010, 03:35 PM
chisigma
It is easy to see that is...

$a_{n+1} = a_{n} \frac{2n+1}{n+1}$ (1)

... so that is...

$\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} = 2$ (2)

... and that means that is...

$\lim_{n \rightarrow \infty} a_{n} = \infty$ (3)

Kind regards

$\chi$ $\sigma$