159k views
4 votes
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

I need help with this practice from my ACT prep guideIt asks to answer (a) and (b-example-1
User Haatschii
by
8.5k points

1 Answer

4 votes

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.

User Edwin
by
8.3k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories