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Assignment 1 (Sections 9.1 and 9.2)

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)