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

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

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

Finding nth Partial Sums

In Exercises 1–6, find a formula for the nth partial sum of each series and use it to find the series’ sum if the series converges.

2 + (2/3) + (2/9) + (2/27) + … + (2 / 3ⁿ⁻¹) + …

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]

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)]

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)

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))