6–9. Determine whether the following sequences converge or diverge, and state whether they are monotonic or whether they oscillate. Give the limit when the sequence converges.


{(−0.7)ⁿ}

40–62. Choose your test Use the test of your choice to determine whether the following series converge.

∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0

Define the remainder of an infinite series.

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 3 to ∞) (2k²) / (k² − k − 2)

21–42. Geometric series Evaluate each geometric series or state that it diverges.  

21.∑ (k = 0 to ∞) (1/4)ᵏ

For what values of r does the sequence {rⁿ} converge? Diverge?

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 2 to ∞) ln((k + 1)k⁻¹) / (ln k × ln(k + 1))