Question

I need help with this practice problem Answer these below as well, The value of r from the ratio test is A. 0B. 0.25C. 1D. 4The series A. Converges B. Diverges

Answer 1

Given the series below

`\sum ^(\infty)_(n\mathop=1)(4^(n+1))/(4n+1)`

By the limit comparison test, the series diverges

Hence, the series diverges

Where

`\begin{gathered} a_n=(4^(n+1))/(4n+1) \\ \text{And } \\ a_(n+1)=(4^(n+1+1))/(4(n+1)+1)=\frac{4^{n+2_{}}}{4n+1+1}=(4^(n+2))/(4n+2) \end{gathered}`

For the ratio test,

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

Substitute for the series

`=((4^(n+2))/(4n+2))*((4n+1)/(4^(n+1)))`

Solve

`\begin{gathered} r=\frac{(4^{n+2-(n+1)_{}})(4n+1)_{}}{4n+2_{}} \\ =((4^(n-n+2-1))(4n+1))/(4n+2) \\ =(4^1(4n+1))/(4n+2) \\ =(16n+4)/(4n+2) \\ =\frac{2^2(4n+1)^{}}{2(n+1)} \\ =(2(4n+1))/(2n+1)\text{ } \\ \lim _(n\rightarrow\infty) \\ =2(2)=4 \end{gathered}`

The value of r from the ratio test is 4

Since r is greater than 1 i.e r > 1

Hence, the series diverges