Question

I need help with this practice problem It asks to solve (a) and (b) Please put these separately so I know which is which ^

Answer 1

Given the series

`\sum ((2n!)/(2^(2n)))`

Let

`a_n=(2n!)/(2^(2n))`

(a) Find r using the using the formula

`r=\lim _(n\to\infty)(a_(n+1))/(a_n)`

Substitute the given values.

`\begin{gathered} r=\lim _(n\to\infty)(2(n+1)!)/(2^(2(n+1)))\cdot(2^(2n))/(2n!) \\ =\lim _(n\to\infty)((n+1))/(2^2)\frac{^{}}{} \\ =\infty \end{gathered}`

(b) By Ratio test, if the value of r is greater than 1, then the series is divergent. Here r > 1. So, the series is divergent.