Question

determine the convergence or divergnece of the sequence with the given nth term. If the sequecne converges find its limit.

a{n}=(1*3*5......**(2n-1))/(2n)^n

Answer 1

So we have
\[\large a_n=\frac{\displaystyle\prod_{k=1}^n(2k-1)}{(2n)^n}=\frac{1\times3\times\cdots\times(2n-1)}{(2n)^n}\]
\[\left\{a_n\right\}_{n=1}^\infty=\left\{\frac{1}{2},\frac{3}{16},\frac{15}{216},\cdots\right\}\]
At first glance, it looks like the sequence is monotonically decreasing to 0, so I'm willing to bet it's convergent.

Are you looking for a more formal approach?

Answer 2

yes, how to solve it anayltically