Which of the following series is/are convergent by the alternating series test? (4 points)
I. the summation from n equals 1 to infinity of the quotient of negative 1 raised the n and the natural log of n
II. the summation from n equals 1 to infinity of the quotient of negative 1 raised the n plus 1 power and the quantity 4 times n squared plus 1
III. the summation from n equals 1 to infinity of the quotient of negative 1 raised the n plus 1 power times the quotient of n squared and n cubed plus 4
A) I only
B) II only
C) I and II only
D) I, II, and III
To determine which of the given series is/are convergent by the alternating series test, we need to check if the series alternates in sign and if the absolute value of each term in the series decreases as n increases.
Let's analyze each series individually:
I. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the power of n and the natural logarithm of n
First, let's check if the series alternates in sign. Notice that the quotient has (-1)^n in the numerator, which changes sign as n increases. Therefore, the series alternates.
Next, we need to check if the absolute value of each term decreases as n increases. The natural logarithm of n is a positive value for all n greater than or equal to 1, so the term itself does not change sign. However, since we have a quotient, we can focus on the numerator.
When n = 1, the numerator is (-1)^1 = -1. As n increases, the numerator alternates between -1 and 1. The absolute value of each term remains constant at 1, so the absolute value of each term does not decrease as n increases.
Based on this analysis, Series I does not meet the criteria of the alternating series test because the absolute value of each term does not decrease. Therefore, Series I is not convergent by the alternating series test.
Now, let's move on to Series II:
II. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the power of n plus 1 and the quantity 4 times n squared plus 1
Again, we need to check if the series alternates in sign. Similar to Series I, the quotient has (-1)^(n+1) in the numerator, which changes sign as n increases. Therefore, the series alternates.
To check if the absolute value of each term decreases, we can analyze the numerator and denominator separately. As n increases, the numerator alternates between -1 and 1, just like in Series I. However, the denominator, 4n^2 + 1, increases as n increases. As a result, the absolute value of each term decreases.
Since Series II satisfies the criteria of the alternating series test (alternates in sign and absolute value of terms decreases), it is convergent by the alternating series test.
Lastly, let's examine Series III:
III. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the power of n plus 1, times the quotient of n squared and n cubed plus 4
Once again, let's check if the series alternates in sign. The numerator has (-1)^(n+1), which changes sign as n increases, so the series alternates.
To determine if the absolute value of each term decreases, we need to analyze the numerator and denominator separately. Similar to Series II, the numerator alternates between -1 and 1 as n increases. However, the denominator, n^3 + 4, increases as n increases. Therefore, the absolute value of each term does not decrease.
Since Series III does not satisfy the criteria of the alternating series test (the absolute value of each term does not decrease), it is not convergent by the alternating series test.
In summary, Series II is the only series convergent by the alternating series test. Therefore, the answer is:
B) II only.