Which one of the following statements is true about the series the series from n equals 1 to infinity of the quotient of negative 1 raised to the nth power and n ? (4 points) Is this A or B? I am a little confused.
A) It is absolutely convergent.
B) It is the alternating harmonic series and is conditionally convergent.
C) It is divergent.
D) Its sum is 0.
B is true, so you know A is false.
all those words! Just write
sum (-1)^n/n
or, if you just have to do it, maybe
sum(n=1..infinity) (-1)^n/n
To determine the convergence of the series, we need to analyze the behavior of the terms as n approaches infinity. The given series is the series from n equals 1 to infinity of the quotient (-1)^n / n.
When n is odd, (-1)^n is equal to -1, and when n is even, (-1)^n is equal to 1. This implies that the terms alternate between positive and negative values.
The series (-1)^n / n is similar to the alternating harmonic series, except with alternating signs. The alternating harmonic series is given by the series from n equals 1 to infinity of (-1)^(n+1)/n, where the signs alternate starting with a positive term.
The alternating harmonic series is conditionally convergent, meaning that when the signs alternate, it converges but not absolutely. In our case, since the signs alternate as well, the series from n equals 1 to infinity of (-1)^n / n is also conditionally convergent.
Therefore, the correct answer is B) It is the alternating harmonic series and is conditionally convergent.
To determine which statement is true about the given series, we can analyze its convergence by applying the alternating series test.
The alternating series test states that if a series has the form Σ(-1)^(n-1) * a_n, where a_n is a positive, decreasing sequence converging to 0, then the series converges.
In this case, the series can be written as Σ(-1)^(n-1) / n. Let's check if it meets the conditions of the alternating series test:
1. Positivity: The terms in the series are positive since the numerator (-1)^(n-1) alternates between -1 and 1. The denominator n is always positive.
2. Decreasing sequence: To determine if the sequence is decreasing, we can compare two consecutive terms. Let's compare a_n and a_(n+1) by considering their ratio:
a_n = 1/n
a_(n+1) = 1/(n+1)
To simplify the comparison, we can compare consecutive terms by finding the ratio:
a_n / a_(n+1) = (1/n) / (1/(n+1)) = (1/n) * ((n+1)/1) = (n+1)/n = 1 + 1/n
Since the ratio (n+1)/n is greater than 1 for all positive integers n, the sequence is indeed decreasing.
3. Converging to 0: As n approaches infinity, the term 1/n goes to 0. Therefore, the sequence converges to 0.
Based on the analysis above, we can conclude that the given series meets the conditions for the alternating series test and hence is convergent.
Now, let's evaluate the other statements given:
A) It is absolutely convergent: The series is not absolutely convergent because it is an alternating series.
B) It is the alternating harmonic series and is conditionally convergent: This statement is correct. The series Σ(-1)^(n-1) / n is referred to as the alternating harmonic series and is conditionally convergent.
C) It is divergent: This statement is incorrect as we have established that the series converges.
D) Its sum is 0: The sum of the alternating harmonic series is not 0. Although the series converges, its sum is not 0.
Therefore, the correct statement is B) It is the alternating harmonic series and is conditionally convergent.