Which of the following series converge?
A. ∑n=1 to infinity of 1/n^2
B. ∑n=1 to infinity of 1/n
C. ∑n=1 to infinity (-1)^n/(√n)
I think the answer is A and C.
correct
To determine which of the given series converge, we can use the convergence tests. Let's analyze each series separately:
A. ∑n=1 to infinity of 1/n^2:
This series is known as the "p-series" with p = 2. The p-series converges if p > 1 and diverges if p <= 1. In this case, p = 2, which is greater than 1. Therefore, this series converges.
B. ∑n=1 to infinity of 1/n:
This is the harmonic series. The harmonic series diverges, meaning it does not converge. This can be proven using other convergence tests, such as the integral test or the limit comparison test.
C. ∑n=1 to infinity of (-1)^n/√n:
This series is an alternating series, as it alternates between positive and negative terms. To determine convergence for alternating series, we can use the alternating series test. According to the alternating series test, if the terms in an alternating series satisfy two conditions: (1) the absolute value of the terms decreases, and (2) the limit of the terms approaches zero, then the series converges. Let's check whether these conditions are met:
(1) The absolute value of the terms is 1/√n, which decreases as n increases.
(2) Taking the limit as n approaches infinity of 1/√n, we find that the limit is indeed zero.
Since both conditions are satisfied, this alternating series converges.
Therefore, the correct answer is A and C. Series A, ∑n=1 to infinity of 1/n^2, and series C, ∑n=1 to infinity of (-1)^n/√n, both converge.