14. Sequences & Series

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{cosn / n}

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{sinn / 2ⁿ}

55–70. More sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(75n⁻¹ / 99ⁿ) + (5ⁿsinn / 8ⁿ)}

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{√(n² + 1) − n} 

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(1 + (2 / n))ⁿ}

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{ⁿ√(e³ⁿ⁺⁴)} 

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

{n¹⁰ / ln20n}

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

{n¹⁰⁰⁰ / 2ⁿ}

Growth rates of sequences

Use Theorem 10.6 to find the limit of the following sequences or state that they diverge.

aₙ = (6ⁿ + 3ⁿ) / (6ⁿ + n¹⁰⁰)

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a.If limₙ→∞aₙ = 1 and limₙ→∞bₙ = 3, then limₙ→∞(bₙ / aₙ) = 3.

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{√((1 + 1 / 2n)ⁿ)}

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.

{(1 / n)¹⁄ⁿ}

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (–1)ⁿ (3n³ + 4n) / (6n³ + 5)

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = (2ⁿ + 5ⁿ⁺¹) / 5ⁿ

12–24. Limits of sequences Evaluate the limit of the sequence or state that it does not exist.

aₙ = 8ⁿ / n!