Telescoping Series
In Exercises 39–44, find a formula for the nth partial sum of the series and use it to determine if the series converges or diverges. If a series converges, find its sum.
∑ (from n = 1 to ∞) [ (1/n) − (1/(n + 1)) ]

Convergence and Divergence

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

aₙ = (1/n) ∫₁ⁿ (1/x) dx

In Exercises 125–134, determine whether the sequence is monotonic, whether it is bounded, and whether it converges.

aₙ = (4ⁿ⁺¹ + 3ⁿ) / 4ⁿ

In Exercises 15–22, determine if the geometric series converges or diverges. If a series converges, find its sum.

1 − (2/e) + (2/e)² − (2/e)³ + (2/e)⁴ − …

Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.

∑ (from n = 1 to ∞) (1 − 1/n)ⁿ

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 1 to ∞) 1 / n⁰·²

Applying the Integral Test

Use the Integral Test to determine if the series in Exercises 1–12 converge or diverge. Be sure to check that the conditions of the Integral Test are satisfied.

∑ (from n = 2 to ∞) ln(n²) / n