Comprehensive Tests for Convergence and Divergence of Infinite Series

Tests for convergence and divergence help determine if an infinite series approaches a finite limit (converges) or grows without bound (diverges). Several primary tests include:

1. Nth-Term Test: If the \( n \)-th term of a series does not approach zero as \( n \to \infty \), the series diverges.
2. Integral Test: Used when terms are positive and decreasing, integrating the function representing the series can show convergence if the integral is finite.
3. Comparison Test: By comparing the series to a known convergent or divergent series, convergence or divergence can be established.
4. Limit Comparison Test: Useful when two series behave similarly; taking the limit of their ratio confirms convergence or divergence.
5. Ratio Test: For series with factorials or exponential terms, the limit of the ratio of consecutive terms indicates convergence if less than \(1\), divergence if greater than \(1\).
6. Root Test: By taking the \( n \)-th root of terms and finding the limit, convergence is indicated if less than \(1\).

Let’s explore each test further:

1. Nth-Term Test: This simplest test checks if the \( n \)-th term of a series approaches zero as \( n \to \infty \). If it does not, the series diverges. However, if it does approach zero, further testing is required since this test alone cannot confirm convergence.
2. Integral Test: For series with positive, continuous, and decreasing terms, integrating a corresponding function, \( f(x) \), can reveal convergence or divergence. If the improper integral \( \int_{1}^{\infty} f(x) \, dx \) is finite, the series converges; if not, it diverges.
3. Comparison Test: Useful when comparing to a known series, this test involves determining whether terms of the series are smaller or larger than a known convergent or divergent series.
4. Limit Comparison Test: Often used when the terms of two series behave similarly, we take the limit of the ratio of their terms. If the limit is a positive, finite constant, both series will either converge or diverge together.
5. Ratio Test: Especially useful for series with factorials or exponentials, this test evaluates \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If the result is less than \(1\), the series converges absolutely; if greater than \(1\), it diverges. A result equal to \(1\) requires other tests for a conclusion.
6. Root Test: This test examines \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). If the limit is less than \(1\), the series converges absolutely; if greater than \(1\), it diverges. Like the ratio test, a result of \(1\) is inconclusive.

Each of these tests offers distinct insights and is applied based on the series’ characteristics. Together, they form a toolkit for analyzing series behavior, integral in fields that rely on infinite summations, such as physics, engineering, and applied mathematics.