Absolute and Conditional Convergence
Which of the series in Exercises 15–48 converge absolutely, which converge, and which diverge? Give reasons for your answers.
∑ (from n = 1 to ∞) [(-1)ⁿ (√(n² + n) − n)]

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First, identify the general term of the series: \(a_n = (-1)^n (\sqrt{n^2 + n} - n)\).

To analyze absolute convergence, consider the absolute value of the terms: \(|a_n| = \sqrt{n^2 + n} - n\).

Simplify \(|a_n|\) by rationalizing or using algebraic manipulation: multiply numerator and denominator by \(\sqrt{n^2 + n} + n\) to get \(|a_n| = \frac{n}{\sqrt{n^2 + n} + n}\).

Examine the limit of \(|a_n|\) as \(n \to \infty\) to determine if the series of absolute values converges. If it does, the series converges absolutely.

If the series does not converge absolutely, check for conditional convergence by applying the Alternating Series Test: verify if \(a_n\) tends to zero and if the sequence \(|a_n|\) is decreasing for sufficiently large \(n\).

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Absolute Convergence

A series ∑aₙ converges absolutely if the series of absolute values ∑|aₙ| converges. Absolute convergence implies convergence regardless of the sign of terms, and it guarantees that rearranging terms does not affect the sum. Testing absolute convergence often involves comparison or limit comparison tests.

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07:51

Choosing a Convergence Test

Conditional Convergence

A series converges conditionally if it converges but does not converge absolutely. This means ∑aₙ converges, but ∑|aₙ| diverges. Conditional convergence often occurs in alternating series where the terms decrease in magnitude and approach zero, as described by the Alternating Series Test.

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07:51

Choosing a Convergence Test

Alternating Series Test

The Alternating Series Test states that an alternating series ∑(-1)ⁿbₙ converges if the sequence {bₙ} is positive, decreasing, and approaches zero as n→∞. This test helps determine conditional convergence when absolute convergence fails, especially for series with terms involving alternating signs.

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