Understanding the Integral Test for Series Convergence
The Integral Test is a powerful method for determining the convergence or divergence of an infinite series. It compares the series to an improper integral of a related function. For a series \(\sum a_n\), if \(f(x)\) is a continuous, positive, decreasing function where \(a_n = f(n)\), the convergence of the integral \(\ \int_1^\infty f(x) \, […]
Exploring the Alternating Series Test: Convergence and Divergence
The Alternating Series Test is used to determine the convergence of series with alternating positive and negative terms. For a series to pass this test, two conditions must be met: the terms of the series must decrease in absolute value, and the terms must approach zero as the series progresses. If these conditions are satisfied, […]