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
I only
II only
I and II only
I, II, and III
The alternating series test states that a series is convergent if the terms alternate in sign and decrease in absolute value. Let's analyze each series to determine if they satisfy this condition.
I. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n and the natural log of n
The terms alternate in sign since we have (-1)^n. However, the natural log of n does not decrease in absolute value as n increases. Therefore, series I does not satisfy the conditions of the alternating series test.
II. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus 1 power and the quantity 4 times n squared plus 1
Again, the terms alternate in sign due to (-1)^(n+1). To check if the terms decrease in absolute value, we can compare consecutive terms.
For n = 1: |(-1)^(1+1)/(4*1^2+1)| = 1/5
For n = 2: |(-1)^(2+1)/(4*2^2+1)| = 1/17
Since 1/17 < 1/5, we can observe that the terms do decrease in absolute value. Therefore, series II satisfies the conditions of the alternating series test.
III. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus 1 power times the quotient of n squared and n cubed plus 4
Again, the terms alternate in sign due to (-1)^(n+1). To check if the terms decrease in absolute value, we can compare consecutive terms.
For n = 1: |(-1)^(1+1)*(1^2/(1^3+4))| = 1/5
For n = 2: |(-1)^(2+1)*(2^2/(2^3+4))| = 4/12
Since 4/12 > 1/5, we can observe that the terms do not decrease in absolute value. Therefore, series III does not satisfy the conditions of the alternating series test.
In conclusion, only series II satisfies the conditions of the alternating series test. Therefore, the correct answer is II only.
To determine whether a series is convergent using the alternating series test, we need to check if two conditions are met:
1. The terms of the series alternate in sign.
2. The absolute value of the terms decreases as n increases.
Let's examine each series provided in the options:
I. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n and the natural log of n
To check the sign alternation, we can see that the terms alternate between positive and negative since (-1)^n alternates between -1 and 1.
Now, let's check if the absolute value of the terms decreases. The natural logarithm function, ln(n), increases as n increases. Since the numerator (-1)^n oscillates between -1 and 1 and the denominator ln(n) increases, the absolute value of the terms is not decreasing. Therefore, this series does not meet the conditions of the alternating series test.
II. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus 1 power and the quantity 4 times n squared plus 1
This series also has terms that alternate in sign due to the (-1)^(n+1) term.
Regarding the decrease of the absolute values, we need to examine the terms. Notice the denominator, 4n^2 + 1, is a quadratic function that increases as n increases. Since the numerator oscillates between -1 and 1 and the denominator increases, the absolute value of the terms does decrease. This series meets the conditions of the alternating series test.
III. the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus 1 power times the quotient of n squared and n cubed plus 4
Once again, we have terms that alternate in sign due to the (-1)^(n+1) term.
To check the decrease of the absolute values, we need to analyze the terms. The numerator, (-1)^(n+1) * n^2, remains bounded as n increases. On the other hand, the denominator, n^3 + 4, increases as n increases.
Since the numerator is bounded and the denominator increases, the absolute value of the terms decreases. This series meets the conditions of the alternating series test.
Now, considering all three series:
I is not convergent by the alternating series test.
II is convergent by the alternating series test.
III is convergent by the alternating series test.
Therefore, the correct answer is:
I, II, and III.