Which Of The Following Series Converge

Which of the Following Series Converge? A Comprehensive Guide

Determining the convergence of a series is a crucial concept in calculus and analysis. This article will explore various methods to determine whether an infinite series converges or diverges, focusing on common tests and providing examples to clarify the process. Understanding convergence is essential for applications ranging from physics and engineering to economics and computer science. We'll cover several key tests and demonstrate their application.

Understanding Convergence and Divergence

Before diving into specific tests, let's define the key terms:

• Convergence: A series converges if the sum of its infinite terms approaches a finite limit. This limit is called the sum of the series.
• Divergence: A series diverges if the sum of its infinite terms does not approach a finite limit. This means the sum either approaches infinity, negative infinity, or oscillates without settling on a specific value.

Several tests can help determine whether a series converges or diverges. The choice of test often depends on the form of the series.

Common Convergence Tests

Here are some of the most frequently used tests to determine series convergence:

1. The nth Term Test (Divergence Test): This is a preliminary test. If the limit of the nth term as n approaches infinity is not zero, then the series diverges. However, if the limit is zero, it doesn't necessarily mean the series converges; further testing is required.

Example: Consider the series Σ (n=1 to ∞) n. The limit of the nth term (n) as n approaches infinity is infinity, not zero. Therefore, this series diverges.

2. The Geometric Series Test: A geometric series has the form Σ (n=0 to ∞) arⁿ, where 'a' is the first term and 'r' is the common ratio. This series converges if |r| < 1, and its sum is a/(1-r). It diverges if |r| ≥ 1.

Example: The series Σ (n=0 to ∞) (1/2)ⁿ is a geometric series with a = 1 and r = 1/2. Since |r| < 1, the series converges.

3. The p-Series Test: A p-series has the form Σ (n=1 to ∞) 1/n^p, where p is a positive constant. This series converges if p > 1 and diverges if p ≤ 1.

Example: The series Σ (n=1 to ∞) 1/n² is a p-series with p = 2. Since p > 1, the series converges. The harmonic series, Σ (n=1 to ∞) 1/n (p=1), diverges.

4. The Integral Test: If f(x) is a positive, continuous, and decreasing function on the interval [1, ∞) such that f(n) = a_n for all n, then the series Σ (n=1 to ∞) a_n converges if and only if the improper integral ∫ (1 to ∞) f(x) dx converges.

Example: Consider the series Σ (n=1 to ∞) 1/n². The function f(x) = 1/x² is positive, continuous, and decreasing on [1, ∞). The integral ∫ (1 to ∞) 1/x² dx converges, therefore the series converges.

5. The Comparison Test: This test compares a given series to a known convergent or divergent series. If 0 ≤ a_n ≤ b_n for all n, and Σ b_n converges, then Σ a_n converges. If 0 ≤ b_n ≤ a_n for all n, and Σ b_n diverges, then Σ a_n diverges.

6. The Limit Comparison Test: This is a refined version of the comparison test. If lim (n→∞) a_n/b_n = c, where c is a positive finite number, then Σ a_n and Σ b_n either both converge or both diverge.

7. The Alternating Series Test: This test applies to alternating series (series with alternating positive and negative terms). An alternating series converges if the absolute value of the terms decreases monotonically to zero.

8. The Ratio Test: This test is particularly useful for series involving factorials. If lim (n→∞) |a_n+1/a_n| = L, then the series converges if L < 1, diverges if L > 1, and the test is inconclusive if L = 1.

9. The Root Test: Similar to the ratio test, this test is useful for series with terms raised to a power of n. If lim (n→∞) |a_n|^1/n = L, then the series converges if L < 1, diverges if L > 1, and the test is inconclusive if L = 1.

Conclusion

Determining the convergence of a series requires careful consideration of the series' form and the application of appropriate tests. By understanding and applying these tests, you can accurately determine whether a given series converges or diverges, a fundamental skill in advanced mathematical analysis. Remember to always check the conditions for each test before applying it. Practicing with diverse examples is crucial for mastering this important concept.