Series Behavior with the n-th Term Test: Divergence Test
The n-th Term Test is a fundamental tool in determining whether a series converges or diverges. It examines the behavior of the terms in a series as \(n\) approaches infinity. Specifically, if the limit of the n-th term \(a_n\) does not approach zero, the series diverges. However, if \(\lim_{n \to \infty}\) a_\(n = 0\), the […]
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) \, […]