# Sequences and series: exercise sheet 1

Exercise.

(a) Find the set of all $x ∈ ℝ ∖ { -3 , -5 }$ such that $x + 1 x + 3 < x + 3 x + 5$.

(b) Find the set of all $x ∈ ℝ ∖ { -3 , -4 }$ such that $x + 1 x + 3 < x + 3 x + 4$.

(c) Find the set of all $x ∈ ℝ ∖ { -3 , -2 }$ such that $x + 1 x + 3 < x + 3 x + 2$.

Give your answers as an interval or a union of disjoint intervals.

Exercise.

(a) Find the set of all $x ∈ ℝ$ such that $| x | > x 2 - 1$.

(b) Find the set of all $x ∈ ℝ$ such that $3 | x | > x 2 + 1$.

Exercise.

Consider the sequence defined by $a n = n n + 1$. Prove that $a n → 1$ as follows.

(a) Find as simple an expression as you can for $| a n - 1 |$. Make sure your expression does not involve absolute value signs.

(b) Supposing $ε > 0$ is given, use your answer to (a) to find the set of $n ∈ ℕ$ such that $0 ⩽ | a n - 1 | < ε$ writing the answer as $A ∪ { n ∈ ℕ : n ⩾ F ( ε ) }$, where $A$ is a finite set of naturals numbers and $F ( ε )$ is an expression involving $ε$ only. (Hint: By conjugating surds, show that $| a n - 1 | < 1 2 n$ for all $n$. You don't have to say what your set $A$ is. Why is it finite? Be sure to use $⩾$ not $>$. If you have ${ n ∈ ℕ : n > F ( ε ) }$ you can replace it with ${ n ∈ ℕ : n ⩾ F ( ε ) + 1 }$ if you change the set $A$.)

(c) Write down a proof that $a n → 1$ starting with Let $ε > 0$ be arbitrary and Let $N = ⌈ F ( ε ) ⌉$ .

Exercise.

Consider the statement X, which is

and the sequence defined by $( a n ) = -1 n + 1 n$. Prove that X holds for this sequence for both $l = 1$ and $l = -1$.

All assertions you make in your answers MUST be supported by proofs.