What is the purpose of taking the absolute value for divergence in infinite sums (series)?

 What is the purpose of taking the absolute value for divergence in infinite sums (series)?👇🏾👇 🏾

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The purpose of taking the absolute value in the context of the convergence of infinite sums (series) is primarily to establish criteria for absolute convergence. Here’s why this is important and the key concepts related to it:

1. Absolute Convergence:

   • An infinite series $\sum {𝑎}_{𝑛}$ is said to be absolutely convergent if the series of the absolute values $\sum \mathrm{\mid }{𝑎}_{𝑛}\mathrm{\mid }$ converges.
   • If $\sum \mathrm{\mid }{𝑎}_{𝑛}\mathrm{\mid }$ converges, then $\sum {𝑎}_{𝑛}$ also converges. This is a powerful criterion because absolute convergence implies regular convergence.

2. Comparison and Ratio Tests:

   • Tests like the Comparison Test, Ratio Test, and Root Test often require working with the absolute values of the terms of the series.
   • For example, in the Ratio Test, we look at ${\lim }_{𝑛\to \mathrm{\infty }}\mid \frac{{𝑎}_{𝑛+1}}{{𝑎}_{𝑛}}\mid$. The absolute value ensures we are comparing the magnitudes of the terms, which is crucial for determining convergence without being affected by alternating signs.

3. Handling Alternating Series:

   • For series with terms that change sign, such as alternating series, absolute convergence helps manage the sign changes.
   • For instance, the Alternating Series Test can determine the convergence of $\sum (-1{)}^{𝑛}{𝑎}_{𝑛}$ where ${𝑎}_{𝑛}\ge 0$. However, if $\sum \mathrm{\mid }(-1{)}^{𝑛}{𝑎}_{𝑛}\mathrm{\mid }=\sum {𝑎}_{𝑛}$ converges, we have absolute convergence, which is a stronger form of convergence.

4. Rearrangement Theorem:

   • The Riemann Series Theorem states that if a series is conditionally convergent (converges, but not absolutely), then its terms can be rearranged to converge to any value or diverge.
   • Absolute convergence prevents this issue, as an absolutely convergent series retains its sum regardless of how the terms are rearranged.

Example Illustrations

1. Alternating Harmonic Series:

   • ${\sum }_{𝑛=1}^{\mathrm{\infty }}\frac{(-1{)}^{𝑛+1}}{𝑛}$ (the alternating harmonic series) converges conditionally but not absolutely.
   • Taking the absolute value gives ${\sum }_{𝑛=1}^{\mathrm{\infty }}\mid \frac{(-1{)}^{𝑛+1}}{𝑛}\mid ={\sum }_{𝑛=1}^{\mathrm{\infty }}\frac{1}{𝑛}$, which diverges.

2. Geometric Series:

   • For a geometric series ${\sum }_{𝑛=0}^{\mathrm{\infty }}𝑎{𝑟}^{𝑛}$ with $\mathrm{\mid }𝑟\mathrm{\mid }<1$, absolute convergence means we look at ${\sum }_{𝑛=0}^{\mathrm{\infty }}\mathrm{\mid }𝑎{𝑟}^{𝑛}\mathrm{\mid }$, which converges if $\mathrm{\mid }𝑟\mathrm{\mid }<1$.

Practical Implications

• Simplification: Working with absolute values often simplifies analysis since absolute values remove concerns about oscillations and sign changes.
• General Convergence: Proving absolute convergence usually suffices to establish convergence of the original series, thus providing a straightforward pathway to demonstrate convergence.
• Stronger Results: Absolute convergence results are stronger and more robust, providing more control over the behavior of the series.

In summary, taking the absolute value for divergence in infinite sums is crucial for establishing absolute convergence, simplifying convergence tests, and ensuring robust convergence properties that aren't sensitive to term rearrangement or sign changes.