Question

find the limit of (n!)/1*3*5*.........(2n-1) as n approaches infinity

Answer 1

\[\lim_{n \rightarrow \infty}\frac{ n! }{ 1\times3\times5\times......(2n-1) }\]

Answer 2

\[\frac{ n! }{ 1\times3\times5\times\cdots\times(2n-1) }\]\[=\frac{ 1\times2\times3\times\cdots\times n }{ 1\times3\times5\times\cdots\times(2n-1) }=\prod_{i=1}^{n}\frac{i}{2i-1}\]

\[=\frac{1}{1}\times\prod_{i=2}^{n}\frac{i}{2i-1}\]
\[\leq\prod_{i=2}^{n}\frac{2}{3}=\left(\frac{2}{3}\right)^{n-1}\to0 \text{ as }n\to\infty\]

Answer 3

I don't know the notation that you are using. What is..\[\prod_{i=1}^{n}\] ?

Answer 4

\[\prod_{i=1}^{n}a_{i}=a_{1}\times a_{2}\times\cdots\times a_{n}\]