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 + √n)]

In Exercises 37–42, find the series’ radius of convergence.

∑ (from n = 1 to ∞) [ (n!)² / (2ⁿ (2n)!) ] xⁿ

Finding a Sequence’s Formula

In Exercises 13–30, find a formula for the nth term of the sequence.

2, 6, 10, 14, 18, …Every other even positive integer

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [4ⁿ / (3n)ⁿ]

Determining Convergence or Divergence

Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.

∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)

Series with Geometric Terms

In Exercises 7–14, write out the first eight terms of each series to show how the series starts. Then find the sum of the series or show that it diverges.

∑ (from n = 0 to ∞) [(1 / 2ⁿ) + ((-1)ⁿ / 5ⁿ)]

Limit Comparison Test

In Exercises 9–16, use the Limit Comparison Test to determine if each series converges or diverges.

∑ (from n=1 to ∞) n(n + 1) / ((n² + 1)(n − 1))