Using The Rearrangement Property Find The Sum

The rearrangement property offers a fascinating and powerful way to find the sum of certain infinite series. It hinges on the crucial distinction between absolutely convergent and conditionally convergent series. Understanding and correctly applying this property can unlock elegant solutions to summation problems that might seem daunting at first glance.

Understanding the Rearrangement Property

The rearrangement property essentially states that the sum of an absolutely convergent series remains the same regardless of how its terms are reordered. However, this is not the case for conditionally convergent series, where a rearrangement can lead to a different sum, or even divergence.

Let's break down these key concepts:

• Infinite Series: An infinite series is the sum of an infinite number of terms. It can be represented as:

  ∑_n=1^∞ a_n = a₁ + a₂ + a₃ + ...

• Convergence: An infinite series converges if its sequence of partial sums approaches a finite limit. The n-th partial sum (S_n) is the sum of the first n terms:

  S_n = a₁ + a₂ + a₃ + ... + a_n

  If lim_n→∞ S_n = L, where L is a finite number, then the series converges to L.

• Absolute Convergence: An infinite series ∑_n=1^∞ a_n is said to be absolutely convergent if the series of the absolute values of its terms, ∑_n=1^∞ |a_n|, converges.

• Conditional Convergence: An infinite series ∑_n=1^∞ a_n is conditionally convergent if it converges, but the series of the absolute values of its terms, ∑_n=1^∞ |a_n|, diverges.

Why Does Rearrangement Matter?

The behavior of rearranging terms in a series boils down to how the positive and negative terms contribute to the sum. In an absolutely convergent series, the positive and negative terms are "balanced" enough that no matter how you rearrange them, the sum will always tend toward the same limit. Think of it like rearranging a finite sum – the order doesn't change the answer.

However, in a conditionally convergent series, the positive and negative terms are individually divergent, but their delicate interplay leads to a convergent sum. By strategically rearranging them, you can manipulate the rate at which the partial sums approach a limit, ultimately affecting the final sum.

The Riemann Rearrangement Theorem: This theorem formalizes the behavior of conditionally convergent series. It states that for any conditionally convergent series and any real number L, there exists a rearrangement of the series that converges to L. Furthermore, there exist rearrangements that diverge to +∞ or -∞. This highlights the dramatic effect that rearrangement can have on conditionally convergent series.

Identifying Absolutely and Conditionally Convergent Series

Before you can apply the rearrangement property, you must determine whether a series is absolutely or conditionally convergent. Here's a rundown of common tests:

Tests for Absolute Convergence (and therefore convergence):

• Comparison Test: If 0 ≤ |a_n| ≤ b_n for all n greater than some integer N, and ∑ b_n converges, then ∑ |a_n| converges (and thus ∑ a_n converges absolutely).
• Ratio Test: Let L = lim_n→∞ |a_n+1 / a_n|.
  • If L < 1, then ∑ |a_n| converges (and thus ∑ a_n converges absolutely).
  • If L > 1, then ∑ |a_n| diverges.
  • If L = 1, the test is inconclusive.
• Root Test: Let L = lim_n→∞ (|a_n|)^1/n.
  • If L < 1, then ∑ |a_n| converges (and thus ∑ a_n converges absolutely).
  • If L > 1, then ∑ |a_n| diverges.
  • If L = 1, the test is inconclusive.
• Integral Test: If f(x) is a continuous, positive, and decreasing function on the interval [1, ∞) and f(n) = a_n, then the series ∑ a_n and the integral ∫₁^∞ f(x) dx either both converge or both diverge. Since we're concerned with absolute convergence, apply the integral test to ∑ |a_n| which corresponds to ∫₁^∞ |f(x)| dx.

Tests for Conditional Convergence (after showing it's not absolutely convergent):

• Alternating Series Test (Leibniz's Test): If the series is of the form ∑ (-1)ⁿ b_n or ∑ (-1)ⁿ⁺¹ b_n, where b_n > 0 for all n, and if:
  • b_n is a decreasing sequence (b_n+1 ≤ b_n for all n)
  • lim_n→∞ b_n = 0 Then the alternating series converges. Important: To conclude conditional convergence, you must also show that ∑ b_n diverges (i.e., the series of absolute values diverges).

Example: The alternating harmonic series ∑_n=1^∞ (-1)ⁿ⁺¹ / n = 1 - 1/2 + 1/3 - 1/4 + ... is a classic example of a conditionally convergent series. The alternating series test shows it converges. However, the harmonic series ∑_n=1^∞ 1/n diverges. Therefore, the alternating harmonic series converges conditionally.

Applying the Rearrangement Property to Find Sums

The rearrangement property, when applicable (i.e., for absolutely convergent series), can be a powerful tool. Here's how to use it:

1. Prove Absolute Convergence: The most crucial step. You must establish that the series converges absolutely using one of the tests described above. If it's conditionally convergent, rearrangement will change the sum, rendering the method invalid.
2. Rearrange the Terms: Strategically rearrange the terms of the series to a form that simplifies the summation. Look for patterns, common denominators, or opportunities to group terms in a way that makes the series easier to evaluate. This is where creativity and problem-solving skills come into play.
3. Evaluate the Rearranged Series: Use known summation formulas, telescoping series techniques, or other methods to find the sum of the rearranged series. Since you've already proven absolute convergence, you know that the sum of the rearranged series is equal to the sum of the original series.

Examples:

Example 1: A Geometric Series in Disguise

Consider the series:

S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

This is a simple geometric series, and we know its sum is 2. Let's see how rearrangement can (trivially, in this case) confirm this.

1. Absolute Convergence: This is a geometric series with a common ratio of 1/2. Since |1/2| < 1, the series converges absolutely.
2. Rearrangement (unnecessary here, but illustrative): We could rearrange it as 1/2 + 1 + 1/8 + 1/4 + 1/16 +... The point is, any rearrangement will still sum to the same value.
3. Evaluate: The sum is a standard geometric series: a / (1 - r) = 1 / (1 - 1/2) = 1 / (1/2) = 2.

Example 2: A More Interesting Rearrangement

Consider the series:

S = ∑_n=1^∞ (1/2ⁿ + 1/3ⁿ) = (1/2 + 1/3) + (1/4 + 1/9) + (1/8 + 1/27) + ...

1. Absolute Convergence: We can split this series into two separate geometric series:

   ∑_n=1^∞ (1/2ⁿ + 1/3ⁿ) = ∑_n=1^∞ (1/2ⁿ) + ∑_n=1^∞ (1/3ⁿ)

   Both of these geometric series converge absolutely (r = 1/2 and r = 1/3, respectively). The sum of absolutely convergent series is also absolutely convergent.

2. Rearrangement: The rearrangement is already implicit in splitting the sum.

3. Evaluate: We can evaluate each geometric series separately:

   ∑_n=1^∞ (1/2ⁿ) = (1/2) / (1 - 1/2) = 1 ∑_n=1^∞ (1/3ⁿ) = (1/3) / (1 - 1/3) = (1/3) / (2/3) = 1/2

   Therefore, S = 1 + 1/2 = 3/2.

Important Note: Attempting to apply this to a conditionally convergent series will lead to incorrect results. Remember the Riemann Rearrangement Theorem! You can make a conditionally convergent series sum to any real number you want by rearranging the terms.

Cautions and Common Mistakes

• Forgetting to Prove Absolute Convergence: This is the biggest mistake. You must rigorously prove absolute convergence before applying the rearrangement property. Otherwise, your result is meaningless.
• Confusing Absolute and Conditional Convergence: Understanding the difference is fundamental. If you mistakenly believe a conditionally convergent series is absolutely convergent, you will get the wrong answer.
• Incorrectly Applying Convergence Tests: Be careful when applying the ratio test, root test, integral test, and other convergence tests. Ensure you meet all the necessary conditions for the test to be valid.
• Algebraic Errors in Rearrangement: Double-check your algebra when rearranging the terms of the series. A simple mistake can lead to an incorrect sum.
• Assuming All Series Converge: Not all infinite series converge. Many diverge to infinity or oscillate. Always determine whether a series converges before attempting to find its sum.

Advanced Applications and Examples

While the basic principle of the rearrangement property is straightforward, its application can become quite sophisticated in more advanced problems.

Example 3: A Series Involving Trigonometric Functions

Consider the series:

S = ∑_n=1^∞ (sin(n) / 2ⁿ) = sin(1)/2 + sin(2)/4 + sin(3)/8 + ...

1. Absolute Convergence: We can use the comparison test. Since |sin(n)| ≤ 1 for all n, we have:

   |sin(n) / 2ⁿ| ≤ 1 / 2ⁿ

   The series ∑_n=1^∞ 1 / 2ⁿ is a convergent geometric series. Therefore, by the comparison test, the series ∑_n=1^∞ |sin(n) / 2ⁿ| converges, and the original series converges absolutely.

2. Rearrangement and Evaluation (requires complex numbers): This requires a clever trick using complex numbers. Recall Euler's formula:

   e^ix = cos(x) + i sin(x)

   Therefore, sin(x) = Im(e^ix), where Im(z) denotes the imaginary part of the complex number z. We can rewrite the series as:

   S = ∑_n=1^∞ Im(eⁱⁿ / 2ⁿ) = Im( ∑_n=1^∞ (eⁱ / 2)ⁿ )

   The series inside the imaginary part is now a geometric series with a common ratio of eⁱ / 2. Since |eⁱ / 2| = 1/2 < 1, this geometric series converges.

   The sum of the geometric series is:

   ∑_n=1^∞ (eⁱ / 2)ⁿ = (eⁱ / 2) / (1 - eⁱ / 2) = eⁱ / (2 - eⁱ)

   Multiplying the numerator and denominator by the conjugate of the denominator (2 - e^-i) gives:

   eⁱ / (2 - eⁱ) * (2 - e^-i) / (2 - e^-i) = (2eⁱ - 1) / (4 - 2eⁱ - 2e^-i + 1) = (2eⁱ - 1) / (5 - 4cos(1))

   Substituting eⁱ = cos(1) + i sin(1):

   (2(cos(1) + i sin(1)) - 1) / (5 - 4cos(1)) = (2cos(1) - 1 + 2i sin(1)) / (5 - 4cos(1))

   Finally, taking the imaginary part:

   S = Im( ∑_n=1^∞ (eⁱ / 2)ⁿ ) = (2sin(1)) / (5 - 4cos(1))

   Therefore, the sum of the original series is (2sin(1)) / (5 - 4cos(1)). This example shows how the rearrangement property, combined with complex numbers, can solve challenging summation problems.

FAQ

• Q: Can I rearrange any convergent series?

  A: No. You can only rearrange absolutely convergent series without changing the sum. Rearranging conditionally convergent series will likely change the sum, or even cause the series to diverge.

• Q: What happens if I rearrange a divergent series?

  A: Rearranging a divergent series typically doesn't make it converge. It will usually remain divergent, although the specific way it diverges might change.

• Q: Is there a way to know what the new sum will be if I rearrange a conditionally convergent series?

  A: The Riemann Rearrangement Theorem states that you can rearrange a conditionally convergent series to converge to any real number, or to diverge to +∞ or -∞. Determining the specific rearrangement needed to achieve a particular sum can be complex and requires careful manipulation of the positive and negative terms.

• Q: Are there any series that are neither absolutely nor conditionally convergent?

  A: Yes. A series can diverge. For example, the series ∑_n=1^∞ (-1)ⁿ oscillates and does not converge (it's divergent).

Conclusion

The rearrangement property is a valuable tool in the analysis of infinite series. However, it is essential to understand the crucial distinction between absolute and conditional convergence. Applying the rearrangement property to absolutely convergent series allows for manipulation and simplification to find sums more easily. Conversely, attempting to rearrange conditionally convergent series without proper consideration will lead to incorrect results, as the Riemann Rearrangement Theorem demonstrates the sensitivity of such series to the order of their terms. By carefully verifying absolute convergence and applying appropriate techniques, you can leverage the rearrangement property to unlock elegant solutions to a wide range of summation problems.