Question

Use the specified test to determine whether the series converges. You have to show firstly that the test is applicable. For instance, in applying the integral test, you have to verify that the function you're integrating is positive and decreasing for all positive values of 'n.' Please provide the details of the test you're using, the series you're evaluating, and the steps to determine its convergence.

Answer 1

To determine the convergence of a series, we will use the Integral Test. This test requires verifying that the function is positive and decreasing for all positive values of n. If the test conditions are met, we can apply the Integral Test to assess the convergence of the series.

Step-by-step explanation:

To determine the convergence of a series, we will use the Integral Test. The Integral Test states that if the function f(x) is positive and decreasing for all positive values of x, and the series is of the form ∑f(n), then the series converges if and only if the integral ∫f(x)dx from 1 to infinity is convergent.

Following the guidelines of the test, let's first verify if the function is positive and decreasing. Suppose the series we are evaluating is ∑a(n), where a(n) is the nth term of the series. We need to check if a(n) is positive and decreasing for all positive values of n.

If a(n) is positive for all n, then the test is applicable. Additionally, if a(n+1) ≤ a(n) for all n, then a(n) is decreasing. If both conditions hold, we can proceed to apply the Integral Test.