Sequences and series: exercise sheet 1

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

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

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

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

(a) Find the set of all x∈ℝ such that x>x2-1.

(b) Find the set of all x∈ℝ such that 3x>x2+1.

Give your answers as an interval or a union of disjoint intervals. Leave any numbers in your answers in surd form.

Consider the sequence defined by an=nn+1. Prove that an→1 as follows.

(a) Find as simple an expression as you can for an-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≤an-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 an-1<12n 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 an→1 starting with Let ε>0 be arbitrary and Let N=⌈F(ε)⌉.

Consider the statement X, which is

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