Question

I need help with this practice problem It asks to answer (a) & (b)

Answer 1

Given series,

$\sum ^{\infty}_{n\mathop=1}(\frac{2n!}{2^{2n}})$

a.The value of r is

$\begin{gathered} r=\lim _{n\to\infty}|\frac{a_{n+1}}{a_n}| \\ \Rightarrow r=\lim _{n\to\infty}\frac{(2n+1)!}{2^{2(n+1)}}\times\frac{2^{2n}}{(2n)!} \\ \Rightarrow r=\lim _{n\to\infty}\frac{2(n+1)}{4}=\infty \end{gathered}$

b. Since r is infinity so the series is divergent.