To apply the alternating series test, we need to check the following two conditions:
1. The terms of the series must alternate in sign.
2. The absolute value of the terms must decrease as n increases.
Let's examine the given series:
a_n = (-1)^n * (3n) / (4n - 1)
First, let's check for the alternating sign.
When n is odd, (-1)^n = -1, so the term becomes:
a_n = (-1) * (3n) / (4n - 1) = -3n / (4n - 1)
When n is even, (-1)^n = 1, so the term becomes:
a_n = (3n) / (4n - 1)
Therefore, the terms of the series do alternate in sign.
Next, let's check the absolute value of the terms.
|a_n| = |(-1)^n * (3n) / (4n - 1)| = |(3n) / (4n - 1)|
When n = 1, |a_n| = |(3 * 1) / (4 * 1 - 1)| = 3/3 = 1
When n = 2, |a_n| = |(3 * 2) / (4 * 2 - 1)| = 6/7
When n = 3, |a_n| = |(3 * 3) / (4 * 3 - 1)| = 9/11
...
As n increases, the numerator (3n) increases, but the denominator (4n - 1) increases faster. So, the absolute value of the terms decreases.
Since the series satisfies both conditions of the alternating series test, we can conclude that the series converges.
Therefore, the answer is B) The series converges.