Which Of The Following Series Is Absolutely Convergent

Absolutely convergent series represent a cornerstone concept in the study of infinite series within mathematical analysis. Understanding which series possess this property is crucial for determining their behavior and applicability in various mathematical contexts. In this comprehensive exploration, we will delve into the definition of absolute convergence, explore a range of series, and analyze their convergence properties to discern which of them are absolutely convergent.

Defining Absolute Convergence

An infinite series ∑an is said to be absolutely convergent if the series of the absolute values of its terms, ∑|an|, converges. In simpler terms, if you take each term in the series, make it positive (or keep it positive if it already is), and the resulting series converges to a finite value, then the original series is absolutely convergent.

• The importance of absolute convergence lies in its implications for the convergence of the original series. If a series is absolutely convergent, it is also convergent in the standard sense. However, the converse is not always true; a series can be convergent without being absolutely convergent. Such series are called conditionally convergent.

Series and Convergence Tests

Before identifying absolutely convergent series, let's explore different types of series and the convergence tests used to determine their behavior.

1. Alternating Series

An alternating series is a series whose terms alternate in sign. It typically has the form ∑(-1)n bn or ∑(-1)n+1 bn, where bn is a positive term.

• Alternating Series Test (Leibniz's Test): An alternating series converges if the sequence of absolute values {bn} is decreasing and converges to zero. That is:

  • bn ≥ 0 for all n
  • bn+1 ≤ bn for all n
  • lim (n→∞) bn = 0

  If these conditions are met, the alternating series converges. However, it does not guarantee absolute convergence.

• Example: Consider the alternating harmonic series:

  ∑ (-1)n+1 / n = 1 - 1/2 + 1/3 - 1/4 + 1/5 - ...

  This series converges by the Alternating Series Test since 1/n is decreasing and approaches zero as n goes to infinity. However, the series of absolute values is the harmonic series, ∑ 1/n, which is divergent. Therefore, the alternating harmonic series is conditionally convergent, not absolutely convergent.

2. Geometric Series

A geometric series has the form ∑ ar^n, where a is a constant and r is the common ratio.

• Convergence: A geometric series converges if |r| < 1 and diverges if |r| ≥ 1.

• Absolute Convergence: A geometric series is absolutely convergent if ∑ |ar^n| = ∑ |a||r|^n converges. This happens when |r| < 1.

• Example: Consider the geometric series:

  ∑ (1/2)^n = 1 + 1/2 + 1/4 + 1/8 + ...

  Here, a = 1 and r = 1/2. Since |r| = 1/2 < 1, the series converges. The series of absolute values is the same, ∑ (1/2)^n, which also converges. Thus, this geometric series is absolutely convergent.

  Now, consider the geometric series:

  ∑ (-1/2)^n = 1 - 1/2 + 1/4 - 1/8 + ...

  Here, a = 1 and r = -1/2. Since |r| = 1/2 < 1, the series converges. The series of absolute values is ∑ |(-1/2)^n| = ∑ (1/2)^n, which also converges. Thus, this geometric series is absolutely convergent.

3. P-Series

A p-series has the form ∑ 1/np, where p is a positive real number.

• Convergence: A p-series converges if p > 1 and diverges if p ≤ 1.

• Absolute Convergence: Since all terms in a p-series are positive, convergence is equivalent to absolute convergence.

• Example: Consider the series:

  ∑ 1/n2 = 1 + 1/4 + 1/9 + 1/16 + ...

  This is a p-series with p = 2. Since p > 1, the series converges. Thus, it is absolutely convergent.

  Now, consider the series:

  ∑ 1/n = 1 + 1/2 + 1/3 + 1/4 + ...

  This is a p-series with p = 1, also known as the harmonic series. Since p ≤ 1, the series diverges. Thus, it is not absolutely convergent.

4. Ratio Test

The Ratio Test is a powerful tool for determining the convergence of a series. It involves taking the limit of the ratio of consecutive terms.

• Ratio Test: For a series ∑ an, let:

  L = lim (n→∞) |an+1 / an|

  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.

• Example: Consider the series:

  ∑ n! / nn

  To apply the Ratio Test, we compute:

  |an+1 / an| = |(n+1)! / (n+1)n+1| / |n! / nn| = (n+1)! nn / (n! (n+1)n+1) = (n+1) nn / (n+1)n+1 = nn / (n+1)n = (n / (n+1))n = (1 / (1 + 1/n))n = 1 / (1 + 1/n)n

  Taking the limit as n → ∞:

  L = lim (n→∞) 1 / (1 + 1/n)n = 1 / e

  Since 1 / e < 1, the series converges absolutely.

5. Root Test

The Root Test is another useful method for determining the convergence of a series.

• Root Test: For a series ∑ an, let:

  L = lim (n→∞) |an|1/n

  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.

• Example: Consider the series:

  ∑ (2n + 3n) / nn

  To apply the Root Test, we compute:

  |an|1/n = ((2n + 3n) / nn)1/n = (2n + 3n)1/n / n

  As n → ∞, (2n + 3n)1/n approaches 3 because 3n dominates 2n for large n. Thus,

  L = lim (n→∞) (2n + 3n)1/n / n = 3 / ∞ = 0

  Since L < 1, the series converges absolutely.

Examples of Absolutely Convergent Series

Let's analyze several series to determine whether they are absolutely convergent.

1. ∑ (-1)n / n2

• This is an alternating series. To check for absolute convergence, we consider the series of absolute values:

  ∑ |(-1)n / n2| = ∑ 1 / n2

  This is a p-series with p = 2, which converges because p > 1. Therefore, the series ∑ (-1)n / n2 is absolutely convergent.

2. ∑ sin(n) / n2

• This series is not alternating, but we can still analyze its absolute convergence. Consider the series of absolute values:

  ∑ |sin(n) / n2|

  Since |sin(n)| ≤ 1 for all n, we have:

  |sin(n) / n2| ≤ 1 / n2

  The series ∑ 1 / n2 converges (as it is a p-series with p = 2). By the Comparison Test, ∑ |sin(n) / n2| also converges. Thus, the series ∑ sin(n) / n2 is absolutely convergent.

3. ∑ (-1)n n / (n2 + 1)

• This is an alternating series. To check for absolute convergence, we consider the series of absolute values:

  ∑ |(-1)n n / (n2 + 1)| = ∑ n / (n2 + 1)

  We can compare this series to ∑ 1 / n using the Limit Comparison Test:

  lim (n→∞) (n / (n2 + 1)) / (1 / n) = lim (n→∞) n2 / (n2 + 1) = 1

  Since the limit is a finite, non-zero number and ∑ 1 / n diverges, ∑ n / (n2 + 1) also diverges. Therefore, the series ∑ (-1)n n / (n2 + 1) is not absolutely convergent.

  However, we can check if the original alternating series converges using the Alternating Series Test. Let bn = n / (n2 + 1). We need to show that bn is decreasing and approaches zero as n goes to infinity.

  • lim (n→∞) n / (n2 + 1) = 0

  • To show that bn is decreasing, consider the derivative of f(x) = x / (x2 + 1):

    f'(x) = (1(x2 + 1) - x(2x)) / (x2 + 1)2 = (1 - x2) / (x2 + 1)2

    For x > 1, f'(x) < 0, which means f(x) is decreasing. Thus, bn is decreasing for n > 1.

  Since the conditions of the Alternating Series Test are met, the series ∑ (-1)n n / (n2 + 1) converges. Therefore, this series is conditionally convergent.

4. ∑ (n / (2n + 1))n

• To check for absolute convergence, we apply the Root Test:

  L = lim (n→∞) |(n / (2n + 1))n|1/n = lim (n→∞) n / (2n + 1) = lim (n→∞) 1 / (2 + 1/n) = 1/2

  Since L < 1, the series converges absolutely.

5. ∑ (-1)n / √n

• This is an alternating series. To check for absolute convergence, we consider the series of absolute values:

  ∑ |(-1)n / √n| = ∑ 1 / √n = ∑ 1 / n1/2

  This is a p-series with p = 1/2, which diverges because p ≤ 1. Therefore, the series ∑ (-1)n / √n is not absolutely convergent.

  However, we can check if the original alternating series converges using the Alternating Series Test. Let bn = 1 / √n. We need to show that bn is decreasing and approaches zero as n goes to infinity.

  • lim (n→∞) 1 / √n = 0
  • The sequence 1 / √n is decreasing.

  Since the conditions of the Alternating Series Test are met, the series ∑ (-1)n / √n converges. Therefore, this series is conditionally convergent.

Summary Table

Here's a summary of the series analyzed and their convergence properties:

Practical Implications

Understanding absolute convergence has several practical implications in mathematical analysis and applications.

1. Rearrangement of Series

One of the most significant properties of absolutely convergent series is that their terms can be rearranged without affecting the sum. This is not true for conditionally convergent series. The Riemann Rearrangement Theorem states that for a conditionally convergent series, the terms can be rearranged to converge to any real number or even diverge.

2. Multiplication of Series

When multiplying two absolutely convergent series, the Cauchy product (a specific way of multiplying the series) converges to the product of the sums of the original series. This property is crucial in various applications, including power series manipulations.

3. Uniform Convergence

Absolute convergence is often related to uniform convergence, which is essential in the study of functions defined by infinite series. If a series of functions converges absolutely, it often implies uniform convergence, which has important implications for the continuity, differentiability, and integrability of the resulting function.

4. Error Estimation

Absolute convergence can simplify error estimation when approximating the sum of an infinite series by a partial sum. The absolute value of the terms provides a bound on the error, making it easier to determine how many terms are needed to achieve a desired level of accuracy.

Conclusion

In summary, absolute convergence is a critical concept in the study of infinite series. It provides a strong condition for the convergence of a series and has important implications for rearranging terms, multiplying series, and ensuring uniform convergence. By understanding and applying various convergence tests, such as the Ratio Test, Root Test, and Comparison Test, we can effectively determine whether a given series is absolutely convergent and leverage its properties in various mathematical applications. Series that are absolutely convergent are generally better behaved and easier to work with than conditionally convergent series, making the concept of absolute convergence an indispensable tool in mathematical analysis.