14. Sequences & Series

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)

13–52. Limits of sequences

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

{√(n² + 1) − n} 

35–44. Limits of sequences Write the terms a₁, a₂, a₃, and a₄ of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. 

{Use of Tech} aₙ₊₁ = (aₙ⁄₁₁ )+ 50;a₀ = 50

13–52. Limits of sequences

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

{(√(4n⁴ + 3n))⁄(8n² + 1)}  

Recursively Defined Sequences

In Exercises 101–108, assume that each sequence converges and find its limit.

a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)

13–52. Limits of sequences

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

{2ⁿ⁺¹3⁻ⁿ}

Find two different explicit formulas for the sequence {1, -2, 3, -4, -5 .....}

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. 

aₙ = 1 + sin(πn / 2)

{Use of Tech} Repeated square roots

Consider the sequence defined by

aₙ₊₁ = √(2 + aₙ),a₀ = √2, for n = 0, 1, 2, 3, …

a.Evaluate the first four terms of the sequence {aₙ}.

State the exact values first, and then the approximate values.

Uniqueness of limits Prove that limits of sequences are unique. That is, show that if L₁ and L₂ are numbers such that aₙ → L₁ and aₙ → L₂, then L₁ = L₂.

72–75. {Use of Tech} Practical sequences

Consider the following situations that generate a sequence

b.Find an explicit formula for the terms of the sequence.

Radioactive decay

A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let Mₙ be the mass of the radioactive material at the end of the nᵗʰ decade, where the initial mass of the material is M₀ = 20g.

Write a recursive formula for the geometric sequence $\(\left\]\lbrace\)18,6,2,\(\frac\)23,\(\ldots\[\right\]\rbrace\)$.

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁. 

aₙ = (−1)ⁿ / 2ⁿ