Use the alternating series test to determine the convergence/divergence of the series the summation from n equals 1 to infinity of the product of negative 1 raised to the nth power and the quotient of 3 times n and the quantity 4 times n minus 1 . (4 points)
A) The alternating series fails.
B) The series converges.
C) The series is conditionally convergent.
D) The summation must start with n = 0 so the alternating series test cannot be applied.
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.
To determine the convergence or divergence of the series using the alternating series test, we need to check two conditions:
1. The terms of the series must alternate sign.
2. The absolute values of the terms must decrease as n increases.
Let's analyze the series step by step.
The given series is:
Σ (-1)^n * (3n) / (4n - 1), where n starts from 1 and goes to infinity.
1. Alternating sign:
The series has (-1)^n as the first term, which alternates the sign with each term. This condition is satisfied.
2. Decreasing absolute values:
To check if the absolute values of the terms decrease, we can consider the ratio of the absolute values of the (n+1)th term and the nth term:
|ratio| = |(-1)^(n+1) * (3(n+1)) / (4(n+1) - 1)| / |(-1)^n * (3n) / (4n - 1)|
Simplifying the ratio:
|ratio| = [(3(n+1)) / (4(n+1) - 1)] / [(3n) / (4n - 1)]
|ratio| = [(3n + 3) / (4n + 3)] / [(3n) / (4n - 1)]
|ratio| = (3n + 3) / (4n + 3) * (4n - 1) / (3n)
Canceling out the common factors in the numerator and denominator:
|ratio| = (3n + 3) / (4n + 3) * (4n - 1) / (3n)
|ratio| = (4n - 1) / (4n + 3)
As n approaches infinity, the ratio reduces to:
|ratio| = (4n - 1) / (4n + 3) ≈ 1
Since the ratio does not converge to zero as n increases, the series does not satisfy the second condition, and the absolute values of the terms do not decrease. Therefore, the series does not satisfy the alternating series test.
So the correct answer is A) The alternating series fails.