Which Of The Following Series Converge

Which of the Following Series Converge? A Deep Dive into Convergence Tests

Determining the convergence or divergence of an infinite series is a fundamental concept in calculus and analysis. Many tests exist to help us navigate this crucial aspect of mathematics, each with its own strengths and limitations. This comprehensive guide will explore several key convergence tests and apply them to various series, providing a detailed understanding of how to determine convergence.

Understanding Convergence and Divergence

Before we delve into specific tests, let's clarify what convergence and divergence mean. An infinite series is simply the sum of an infinite number of terms: ∑_n=1^∞ a_n = a₁ + a₂ + a₃ + ...

• Convergence: A series converges if the sum of its terms approaches a finite limit as the number of terms approaches infinity. In simpler terms, the sum settles down to a specific number.

• Divergence: A series diverges if the sum of its terms does not approach a finite limit. This means the sum either grows infinitely large (positive or negative infinity) or oscillates without settling on a value.

Essential Convergence Tests

Several tests can help determine whether a series converges or diverges. Let's examine some of the most commonly used:

1. The nth Term Test (Divergence Test)

This is the simplest test. If the limit of the nth term as n approaches infinity is not zero (i.e., lim_n→∞ a_n ≠ 0), the series diverges. However, if the limit is zero, the test is inconclusive; the series may converge or diverge. This test only tells us about divergence; it cannot confirm convergence.

Example: Consider the series ∑_n=1^∞ (n+1)/n. Since lim_n→∞ (n+1)/n = 1 ≠ 0, the series diverges.

2. The Integral Test

If the terms of a series are positive, decreasing, and can be represented as a function f(x) that is continuous, positive, and decreasing for x ≥ 1, then the series ∑_n=1^∞ a_n converges if and only if the improper integral ∫₁^∞ f(x) dx converges.

Example: Consider the p-series ∑_n=1^∞ 1/n^p. The corresponding integral is ∫₁^∞ 1/x^p dx. This integral converges if p > 1 and diverges if p ≤ 1. Therefore, the p-series converges if p > 1 and diverges if p ≤ 1.

3. The Comparison Test

This test compares a given series to another series whose convergence or divergence is already known.

• Direct Comparison Test: If 0 ≤ a_n ≤ b_n for all n, and ∑_n=1^∞ b_n converges, then ∑_n=1^∞ a_n also converges. Conversely, if 0 ≤ b_n ≤ a_n for all n, and ∑_n=1^∞ b_n diverges, then ∑_n=1^∞ a_n also diverges.

• Limit Comparison Test: If a_n and b_n are positive terms, and lim_n→∞ (a_n/b_n) = L, where L is a finite positive number, then ∑_n=1^∞ a_n and ∑_n=1^∞ b_n either both converge or both diverge.

Example: Consider the series ∑_n=1^∞ 1/(n² + 1). We can compare this to the convergent p-series ∑_n=1^∞ 1/n² (p=2 > 1). Using the limit comparison test: lim_n→∞ [(1/(n² + 1))/(1/n²)] = 1. Since the limit is a finite positive number and ∑_n=1^∞ 1/n² converges, ∑_n=1^∞ 1/(n² + 1) also converges.

4. The Ratio Test

This test is particularly useful for series involving factorials or exponentials. For a series ∑_n=1^∞ a_n with positive terms:

• If lim_n→∞ |a_n+1/a_n| < 1, the series converges absolutely.
• If lim_n→∞ |a_n+1/a_n| > 1, the series diverges.
• If lim_n→∞ |a_n+1/a_n| = 1, the test is inconclusive.

Example: Consider the series ∑_n=1^∞ (n!)/nⁿ. Applying the ratio test: lim_n→∞ |[(n+1)!/(n+1)ⁿ⁺¹]/[(n!)/nⁿ]| = lim_n→∞ [(n+1)!nⁿ]/[(n+1)ⁿ⁺¹n!] = lim_n→∞ nⁿ/[(n+1)ⁿ] = lim_n→∞ [n/(n+1)]ⁿ = 1/e < 1. Therefore, the series converges.

5. The Root Test

Similar to the ratio test, this test is effective for series with terms involving roots or powers. For a series ∑_n=1^∞ a_n:

• If lim_n→∞ |a_n|^1/n < 1, the series converges absolutely.
• If lim_n→∞ |a_n|^1/n > 1, the series diverges.
• If lim_n→∞ |a_n|^1/n = 1, the test is inconclusive.

Example: Consider the series ∑_n=1^∞ (1/2)ⁿ. Applying the root test: lim_n→∞ |(1/2)ⁿ|^1/n = 1/2 < 1. Therefore, the series converges. (This is a geometric series with r = 1/2, and since |r| < 1, it converges).

6. Alternating Series Test

This test applies specifically to alternating series, which have terms that alternate in sign: ∑_n=1^∞ (-1)ⁿ⁺¹ b_n, where b_n ≥ 0 for all n. The series converges if:

• b_n is decreasing (b_n+1 ≤ b_n for all n)
• lim_n→∞ b_n = 0

Example: The alternating harmonic series ∑_n=1^∞ (-1)ⁿ⁺¹ (1/n) converges because 1/n is decreasing and lim_n→∞ (1/n) = 0.

Applying the Tests: Example Series

Let's analyze the convergence of several series using the tests discussed above:

Series 1: ∑_n=1^∞ (1/n³)

This is a p-series with p = 3 > 1. Therefore, by the integral test (or simply recognizing it as a convergent p-series), this series converges.

Series 2: ∑_n=1^∞ (n² + 1)/(n³ - 1)

Using the limit comparison test, we can compare this to the series ∑_n=1^∞ (1/n). The limit of the ratio is 1, and since ∑_n=1^∞ (1/n) (the harmonic series) diverges, this series also diverges.

Series 3: ∑_n=1^∞ (2ⁿ)/(n!)

Applying the ratio test: lim_n→∞ |(2ⁿ⁺¹)/[(n+1)!] * (n!)/(2ⁿ)| = lim_n→∞ 2/(n+1) = 0 < 1. Therefore, this series converges.

Series 4: ∑_n=1^∞ (-1)ⁿ/(√n)

This is an alternating series. The terms 1/√n are decreasing, and lim_n→∞ (1/√n) = 0. Therefore, by the alternating series test, this series converges.

Series 5: ∑_n=1^∞ (n!)²/(2n)!

Using the ratio test: lim_n→∞ |[(n+1)!]²/[(2n+2)!] * [(2n)!]/[(n!)²]| = lim_n→∞ (n+1)²/[(2n+2)(2n+1)] = 1/4 < 1. The series converges.

Series 6: ∑_n=1^∞ (1/n^0.5)

This is a p-series with p = 0.5 ≤ 1. Therefore, by the integral test, it diverges.

Conclusion

Determining the convergence or divergence of infinite series is a critical skill in advanced mathematics. Understanding and applying the various convergence tests discussed above – the nth term test, integral test, comparison tests, ratio test, root test, and alternating series test – provides a powerful toolkit for analyzing the behavior of infinite series. Remember to carefully consider the characteristics of the series and choose the most appropriate test for a given situation. By systematically applying these tests and understanding their limitations, you can accurately determine whether a series converges or diverges. Practice is key to mastering these techniques.