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)|(a_(n+1))/(a_n)|`

We are given the series:

`\sum ^(\infty)_(n\mathop=1)\mleft((2n!)/(2^(2n))\mright)`

The term an is:

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

And the term an+1 is

`a_(n+1)=(2(n+1)!)/(2^(2n+2))`

Substituting in the limit:

`L=\lim _(n\to\infty)((2(n+1)!)/(2^(2n+2)))/((2n!)/(2^(2n)))`

Operating:

`L=\lim _(n\to\infty)(2(n+1)!\cdot2^(2n))/(2n!\cdot2^(2n+2))`

Simplifying:

`\begin{gathered} L=\lim _(n\to\infty)(2(n+1)\cdot n!\cdot2^(2n))/(2n!\cdot2^(2n)\cdot2^2) \\ L=\lim _(n\to\infty)((n+1))/(2^2) \\ L=\lim _(n\to\infty)((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