Sequences and series: exercise sheet 1

Exercise.

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

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

(c) Find the set of all x { -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 such that | x | > x 2 - 1 .

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

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

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 such that 0 | a n - 1 | < ε writing the answer as A { 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 > F ( ε ) } you can replace it with { 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

ε > 0   n   0 < | a n - l | | a n - l | < ε

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.