Exam 2

Write your name and student number on the front of your blue book together with a grading table. Start each question on a new page. Make sure you SHOW ALL WORK. No points will be given for answers where no justification is given.

1. (21 points)
   1. Evaluate the integral

      

   2. Determine whether the following integrals converge or diverge.

      

2. (24 points) Which of these series converge?

   

3. (20 points) For what values of x do the following i. converge absolutely, ii. converge conditionally, and iii. diverge?

   

4. (15 points) Consider the series

   

   which has nth term and nth partial sum . For each of the following write TRUE or FALSE. If a statement is true, justify your answer by referring to standard theorems, tests, or definitions. If it is false, show that it is false by providing a counter-example and an explanation of why it is a counter-example.

   

5. (20 points)

   1. Using series, find an interval (of x-values) on which can be replaced by with an error less than .
   2. The series

      

      converges to a function for |x|<1. Find the Maclaurin series converging to

      

      for |x|<1. From the form of its Maclaurin series, give a simple formula for as an explicit function of x (not a series). Hence give a simple formula for .

is an example of a continued fraction. The value of z is the limit of the sequence where

(you may assume this sequence converges).

1. Write down a recursive definition for the sequence .
2. Find an exact value for z, the limit of this sequence.