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⁻¹ / (n² + 2n + 1)]

In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.

∑ (from n = 0 to ∞) [(x + 1)²ⁿ] / 9ⁿ

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 8^(1/n)

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (2n + 2)! / (2n − 1)!

Suppose that aₙ > 0 and limₙ→∞ n²aₙ = 0. Prove that ∑aₙ converges.

In Exercises 57–82, use any method to determine whether the series converges or diverges. Give reasons for your answer.

∑ (from n = 1 to ∞) [3ⁿ / n³]

Error Estimation

In Exercises 49–52, estimate the magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series.

1 / (1 + t) = ∑ (from n = 0 to ∞) [(-1)ⁿ tⁿ],0 < t < 1