Which of the following series converges conditionally? (4 points)
A) the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus one power and n cubed
B) the summation from n equals 2 to infinity of the quotient of negative 1 raised to the n plus one power and the natural log of n
C) the summation from n equals 1 to infinity of negative 1 raised to the n power
D) the summation from n equals 1 to infinity of the quotient of the cosine of n times pi over 3 and n factorial
To determine which of the series converges conditionally, we can use the alternating series test. The alternating series test states that if a series follows the pattern of alternating signs and the terms decrease in magnitude, then the series converges.
Let's analyze each series using this test:
A) The series in option A follows the alternating pattern of signs since the quotient of (-1)^(n+1) changes sign as n increases. Also, the terms are in decreasing order because n^3 is increasing faster than n^1. Therefore, option A satisfies the conditions of the alternating series test.
B) The series in option B also follows the alternating pattern of signs. However, to determine if the terms decrease in magnitude, we need to evaluate the ratio of consecutive terms. Let's calculate the ratio:
|r((n+1))/r(n)| = (|-1|^(n+2))/(|-1|^(n+1) * ln(n+1) / ln(n))
Simplifying this expression, we have |ln(n)/(ln(n+1)|.
Since ln(x) is a monotonically increasing function, ln(n) < ln(n+1), and thus |ln(n)/(ln(n+1))| > 1. Therefore, the terms do not decrease in magnitude, which means option B does not satisfy the conditions of the alternating series test.
C) The series in option C is a simple geometric series with alternating signs. The absolute value of each term is constant, 1, so the terms do not decrease in magnitude. Therefore, option C does not satisfy the conditions of the alternating series test.
D) The series in option D has alternating signs given by the cosine function, and the magnitude of the terms decreases as n increases because the factorial term dominates the denominator. This means option D satisfies the conditions of the alternating series test.
Therefore, the series that converges conditionally is option D.