# Sequences and series: exercise sheet 1

Exercise.

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

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

(c) Find the set of all -3,-2 such that +1 +3 +3 +2 .

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

Exercise.

(a) Find the set of all such that 2-1 .

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

Exercise.

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

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

(b) Supposing 0 is given, use your answer to (a) to find the set of such that 0 -1 writing the answer as () , where is a finite set of naturals numbers and () is an expression involving only. (Hint: By conjugating surds, show that -1 12 for all . You don't have to say what your set is. Why is it finite? Be sure to use not . If you have () you can replace it with ()+1 if you change the set .)

(c) Write down a proof that 1 starting with Let 0 be arbitrary and Let = () .

Exercise.

Consider the statement X, which is

0 0 - -

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

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