Question

Determine which of the following series meet the hypothesis of the integral test and if so, use the integral test to prove convergence or divergence of the series.

A) ∑ (n = 1 to [infinity]) (1/n^2)
B) ∑ (n = 1 to [infinity]) (1/n)
C) ∑ (n = 2 to [infinity]) (1/(n(ln(n))^2))
D) ∑ (n = 1 to [infinity]) (1/(n^3 + 1))

Answer 1

The given series are analyzed using the integral test to determine their convergence or divergence.

Step-by-step explanation:

To determine which of the given series meet the hypothesis of the integral test, we need to check if the series satisfy the three criteria: the terms are positive, decreasing, and approach zero as n approaches infinity.

1. A) ∑ (n = 1 to [infinity]) (1/n^2): This series meets the hypothesis of the integral test because the terms are positive, decreasing, and approach zero. To prove convergence, we can use the integral test by comparing it with the integral ∫(1/x^2) dx. Solving this integral, we get ∫(1/x^2) dx = -1/x. Evaluating the integral from 1 to infinity, we have -1/∞ - (-1/1) = 1. Since the integral is finite, the series converges.
2. B) ∑ (n = 1 to [infinity]) (1/n): This series does not meet the hypothesis of the integral test because the terms do not approach zero. The series diverges.
3. C) ∑ (n = 2 to [infinity]) (1/(n(ln(n))^2)): This series meets the hypothesis of the integral test. To prove convergence, we can use the integral test by comparing it with the integral ∫(1/(x(ln(x))^2)) dx. Solving this integral is a bit complex, but we can use the fact that for large values of x, the function 1/(x(ln(x))^2) is decreasing and approaches zero. Therefore, the integral is finite and the series converges.
4. D) ∑ (n = 1 to [infinity]) (1/(n^3 + 1)): This series does not meet the hypothesis of the integral test because the terms do not approach zero. The series diverges.