1. [6 marks] Show that by using the definition of a convergent sequence and no other properties of sequences.
2. [7 marks] Consider the sequence defined by and .
(a) Compute .
(b) Use mathematical induction to show that the sequence is bounded from above (Hint: show that , say.)
(c) (Optional) Use mathematical induction to show that the sequence increases.
(d) Use (b) and (c) above and refer to a theorem given in class (and in the textbook) to conclude that converges.
(e) Find (Hint: see how we have done that part in the similar examples done in class.)
3. [6 marks] Use what was covered in section 9.1 to evaluate the following limits.
(a)
(b)
4. [6 marks] Find the sum if the series converges; otherwise show it diverges.
(a)
(b)
(c)