Question

Use the integral test to determine the convergence or divergence of the series, where n is a positive integer:

a) Determine convergence or divergence based on the integral test
b) Apply the integral test to evaluate the series
c) Find the limit of the series using the integral test
d) Determine the value of n for convergence

Answer 1

To determine the convergence or divergence of a series using the integral test, check if the function is positive, continuous, and decreasing. If it is, evaluate the definite integral. If the integral converges, the series converges. If the integral diverges, the series diverges. The limit of the series determines convergence, and the value of n for convergence can be found by evaluating the integral up to n.

Step-by-step explanation:

To determine the convergence or divergence of the series using the integral test, follow these steps:

1. Let f(n) represent the function of the series.
2. Check if f(n) is positive, continuous, and decreasing for n ≥ 1.
3. If f(n) is positive, continuous, and decreasing for n ≥ 1, then evaluate the definite integral ∫(1 to ∞) f(x) dx.
4. a) If the integral converges, then the series also converges.
5. b) If the integral diverges, then the series also diverges.

For part c, finding the limit of the series using the integral test involves taking the limit as n approaches infinity of the function f(n) used in the integral test. If the limit is finite and nonzero, then the series diverges. If the limit is zero, then the series may converge or diverge.

In part d, the value of n for convergence can be determined by finding the range of n values for which the integral ∫(1 to n) f(x) dx converges.