Question

We will determine whether the series Σ_{n}=Σ_{n=1} {3ⁿ +1}{n{3} / {2}} 3ⁿ +4} is

A convergent
B divergent

Answer 1

To determine whether the series Σ_{n}=Σ_{n=1} {3ⁿ +1}{n{3} / {2}} 3ⁿ +4} is convergent or divergent, we can use the ratio test.

Step-by-step explanation:

To determine whether the series Σ_{n}=Σ_{n=1} {3ⁿ +1}{n{3} / {2}} 3ⁿ +4} is convergent or divergent, we can use the ratio test. The ratio test states that if the limit of the absolute value of the ratio between consecutive terms of a series is less than 1, then the series is convergent. Let's apply the ratio test to this series.

First, let's find the ratio between consecutive terms:

(3^(n+1) + 1)(n(3/2) / 3ⁿ + 4) / ((3ⁿ + 1)(n(3/2) / 3ⁿ + 4))

Simplifying, we get:

3(n+1)(3/2) / (n(3/2)) = 3(3/2)

The ratio is a constant value, which is greater than 1. Therefore, the limit of the ratio does not approach 0, and the series is divergent.