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

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)]

Determining Convergence of Sequences

Which of the sequences whose nth terms appear in Exercises 1–18 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = 1 + (0.9)ⁿ

Assume that bₙ is a sequence of positive numbers converging to 1/3. Determine if the following series converge or diverge.

a. ∑ (from n = 1 to ∞) [(bₙ₊₁ + bₙ) / n 4ⁿ]

Does the series

∑ (from n=1 to ∞) (1/n − 1/n²)

converge or diverge? Justify your answer.

If ∑aₙ is a convergent series of positive terms, prove that ∑sin(aₙ) converges.

In Exercises 67–72, use the results of Exercises 63 and 64 to determine if each series converges or diverges.

∑(from n=2 to ∞) [(ln n)¹⁰⁰⁰ / n¹.⁰⁰¹]

Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 

${\sum }_{n=1}^{\infty }\sin \left(\frac{n\pi }{2}\right)$

Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 

${\sum }_{n=1}^{\infty }\frac{{10}^n}{n!}$