Which of the following series is absolutely convergent
Introduction
In the study of infinite series, absolute convergence is a central concept that distinguishes series that behave well under summation from those that merely converge conditionally. Consider this: an infinite series Σ aₙ is said to be absolutely convergent if the series of absolute values Σ |aₙ| converges. This property guarantees that the original series converges, regardless of the signs of its terms, and it allows the manipulation of series (rearrangement, term‑by‑term operations) without fear of altering the sum.
When faced with a list of candidate series—such as the harmonic series, an alternating harmonic series, a p‑series, a factorial‑based series, and a power‑over‑factorial series—the task is to determine which of the following series is absolutely convergent. Now, the answer hinges on applying convergence tests to each term and examining the behavior of Σ |aₙ|. The following sections outline a systematic approach, provide a scientific explanation of why certain series meet the criterion, and address frequently asked questions to solidify understanding That's the whole idea..
Steps
To identify the absolutely convergent series, follow these steps:
- List the series and write the general term aₙ for each.
- Form the absolute‑value series Σ |aₙ|.
- Apply a suitable convergence test to Σ |aₙ|. Common tests include:
- Comparison Test: compare with a known convergent or divergent series.
- Limit Comparison Test: compute the limit of the ratio of terms.
- Ratio Test: evaluate L = limₙ→∞ |aₙ₊₁ / aₙ|. If L < 1, the series converges absolutely.
- Root Test: evaluate L = limₙ→∞ √[n]{|aₙ|}. If L < 1, absolute convergence follows.
- Interpret the result: if Σ |aₙ| converges, the original series is absolutely convergent; otherwise, it may be only conditionally convergent or divergent.
Applying these steps to each candidate series will reveal which one(s) satisfy absolute convergence Easy to understand, harder to ignore. Turns out it matters..
Scientific Explanation
Understanding Absolute Convergence
Absolute convergence is stronger than ordinary convergence. A series may converge even though the sum of the absolute values diverges (e.g., the alternating harmonic series Σ (-1)ⁿ⁺¹ / n). In such cases, the positive and negative terms cancel each other, leading
