Question

I need help with this practice problem from my prep guide

Answer 1

Ratio Test

The ratio test is a procedure that can help to determine if an infinite series converges or diverges.

The test comes in form of a limit:

$L=\lim _{n\to\infty}|\frac{a_{n+1}}{a_n}|$

We are given the series:

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

The term an is:

$a_n=\frac{2n!}{2^{2n}}$

And the term an+1 is

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

Substituting in the limit:

$L=\lim _{n\to\infty}\frac{\frac{2(n+1)!}{2^{2n+2}}}{\frac{2n!}{2^{2n}}}$

Operating:

$L=\lim _{n\to\infty}\frac{2(n+1)!\cdot2^{2n}}{2n!\cdot2^{2n+2}}$

Simplifying:

$\begin{gathered} L=\lim _{n\to\infty}\frac{2(n+1)\cdot n!\cdot2^{2n}}{2n!\cdot2^{2n}\cdot2^2} \\ L=\lim _{n\to\infty}\frac{(n+1)}{2^2} \\ L=\lim _{n\to\infty}\frac{(n+1)}{4} \end{gathered}$

This limit does not exist since it tends to infinity when n tends to infinity.

(a) The limit does not exist, thus the ratio cannot be evaluated

(b) The series is divergent