Question

I need help with this practice from my ACT prep guideIt asks to answer (a) and (b)But please put these separately so I can see which one is which

Answer 1

a)

The ratio test states that

`\lim _(n\to\infty)\mleft|(a_(n+1))/(a_n)\mright|=L`

If

• L > 1, the series diverges

• L < 1, the series converges absolutely

• L = 1, inconclusive.

Then let's apply the radio test:

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

Then

`\lim _(n\to\infty)\mleft|(a_(n+1))/(a_n)\mright|=\lim _(n\to\infty)(2(n+1)!)/(2^(2(n+1)))\cdot(2^(2n))/(2n!)`

Now we must simplify the expression

`\begin{gathered} \lim _(n\to\infty)\mleft|(a_(n+1))/(a_n)\mright|=\lim _(n\to\infty)(2(n+1)!)/(2n!)\cdot(2^(2n))/(2^(2(n+1))) \\ \\ \lim _(n\to\infty)|(a_(n+1))/(a_n)|=\lim _(n\to\infty)((n+1)!)/(n!)\cdot(2^(2n))/(2^(2n)\cdot2^2) \\ \\ \lim _(n\to\infty)|(a_(n+1))/(a_n)|=\lim _(n\to\infty)(n+1)/(4) \end{gathered}`

Then, we have

`\lim _(n\to\infty)\mleft|(a_(n+1))/(a_n)\mright|=\lim _(n\to\infty)(n+1)/(4)=+\infty`

The value of r from the ratio test is

`r=+\infty`

b)

If r > 1, then we can say that the series diverges.