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.

Recursively Defined Sequences

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

a₁ = 5,aₙ₊₁ = √(5aₙ)

In Exercises 121–124, determine whether the sequence is monotonic and whether it is bounded.

aₙ = 2ⁿ 3ⁿ / n!

Is it true that a sequence {aₙ} of positive numbers must converge if it is bounded above? Give reasons for your answer.

Recursively Defined Sequences

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

a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)

A sequence of rational numbers is described as follows:

1/1,3/2,7/5,17/12,…,a/b,(a + 2b)/(a + b),…

Here the numerators form one sequence, the denominators form a second sequence, and their ratios form a third sequence. Let xₙ and yₙ be, respectively, the numerator and the denominator of the nᵗʰ fraction rₙ = xₙ / yₙ.

b. The fractions rₙ = xₙ / yₙ approach a limit as n increases. What is that limit? (Hint: Use part (a) to show that rₙ² − 2 = ±(1 / yₙ)² and that yₙ is not less than n.)

The first 4 terms of a sequence are $\(\left\[\lbrace\]\sqrt\)3,2\(\sqrt\)3,3\(\sqrt\)3,4\(\sqrt\)3,\(\ldots\[\right\]\rbrace\)$. Continuing this pattern, find the $7^{\(\th\)}$ term.

Determine the first 3 terms of the sequence given by the general formula

$a_{n}=\(\frac{1}{n!+1}\)$

Write the first 6 terms of the sequence given by the recursive formula $a_{n}=a_{n-2}+a_{n-1}$ ; $a_1=1$ ; $a_2=1$.