Answer:
[tex]r=\dfrac{12}{7}[/tex]
series diverges
Step-by-step explanation:
To find the common ratio (r) of a geometric series, divide the (n+1)th term by the nth term.
When n = 1:
[tex]a_1=\dfrac{3^{1+1}}{3(1)+1}=\dfrac{3^2}{4}=\dfrac94[/tex]
When n =2:
[tex]a_2=\dfrac{3^{2+1}}{3(2)+1}=\dfrac{3^3}{7}=\dfrac{27}{7}[/tex]
Therefore,
[tex]r=\dfrac{a_2}{a_1}=\dfrac{\frac{27}{7}}{\frac{9}{4}}=\dfrac{12}{7}[/tex]
A series that converges has a finite limit. If |r| < 1, then the series will converge.
A series that diverges means either the partial sums have no limit or approach infinity. If |r| > 1 then the series diverges.
Therefore, as the limit of the series approaches infinity and it's r value is greater than 1, the series diverges.
