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.

Using the Root Test

In Exercises 9–16, use the Root Test to determine if each series converges absolutely or diverges.

∑(from n=1 to ∞) [(-1)ⁿ (1 − 1/n)ⁿ^²]

(Hint: lim (n→∞) (1 + x/n)ⁿ = eˣ)

Recursively Defined Sequences

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

a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = nπ cos(nπ)

Use power series operations to find the Taylor series at x = 0 for the functions in Exercises 13–30.

sin x – x + (x³ / 3!)

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (n + 3) / (n² + 5n + 6)

Find the value of b for which

1 + eᵇ + e²ᵇ + e³ᵇ + ⋯ = 9.