14. Sequences & Series

Write a formula for the general or $n^{\th}$ term of the geometric sequence where $a_7=1458$ and $r=-3$.

Explain why or why not

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

e.The sequence aₙ = n² / (n² + 1) converge.

a.Does the sequence { k/(k + 1) } converge? Why or why not?

Suppose the sequence { aₙ} is defined by the explicit formula aₙ = 1/n, for n=1, 2, 3, .....Write out the first five terms of the sequence.

Suppose the sequence { aₙ} is defined by the recurrence relation a₍ₙ₊₁₎ = n · aₙ , for n=1, 2, 3 ...., where a₁ = 1. Write out the first five terms of the sequence.

The terms of a sequence of partial sums are defined by Sₙ = ∑ⁿₖ₌₁ k² , for n=1, 2, 3, .....Evaluate the first four terms of the sequence.

Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k. 

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

aₙ = 1/10ⁿ

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

aₙ = (−1)ⁿ / 2ⁿ

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

aₙ = (2ⁿ⁺¹) / (2ⁿ + 1)

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

aₙ = 1 + sin(πn / 2)

21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

aₙ₊₁ = 2aₙ; a₁ = 2

"21–26. Recurrence relations Write the first four terms of the sequence {aₙ} defined by the following recurrence relations.

aₙ₊₁ = 3aₙ-12; a₁ = 10

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. 

aₙ = 1⁄10ⁿ; n = 1, 2, 3, …

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. 

aₙ = 3 + cos(π*ⁿ) ; n = 1, 2, 3, …