Math 252 Online   Graded Assignment #3                                                              

     It is essential to show all steps by hand.  Also, if a method is prescribed, use only that method (no credit otherwise.)

                       Answers must be exact.  Decimal approximations may not be given unless they are asked for.

1.  (3 points each)  Let  be the sequence whose nth term is  .

(i)  Does  converge or diverge?  Explain.  If  converges, determine its limit.

(ii)  Does  converge or diverge?  Explain.  If  converges, determine its sum.

 

2.  (3 points each)  Let  be the sequence whose nth term is  .

(i)  Does  converge or diverge?  Explain.  If  converges, determine its limit.

(ii)  Does  converge or diverge?  Explain.  If  converges, determine its sum.

 

3.  (3 points each)  Let  be the sequence whose nth term is  .

(i)  Does  converge or diverge?  Explain.  If  converges, determine its limit.

(ii)  Does  converge or diverge?  Explain.  If  converges, determine its sum.

4.  (6 points)  For what value(s) of q does the following series converge? 

 

5.  (6 points)  For what value(s) of s does the following series converge? 

 

6.  (4 points each)  Determine whether each of the following series converges or diverges.  Sufficient explanation must be presented.  If a series converges, you do not need to find its sum.  Note that e and  are the well-known irrational numbers.

(i)

 

(ii)

 

(iii)

 

(iv)

 

(v)

7.  (6 points)  Write the following convergent series in sigma notation and compute its sum. 

           

 

8.  (6 points)  Use the Ratio Test to determine whether the series  converges or diverges.

 

 

9.  (4 points)  Use the Limit Comparison Test to show that the series  converges.

 

 

10.  (10 points)  Determine the radius of convergence and the interval of convergence (make sure you clearly label which is which) for the power series   .

 

 

11.  (8 points)  Determine a geometric power series for  centered at c = 2, and determine the interval of convergence.

 

 

 

12.  (8 points)  Let T(x) be the third Taylor polynomial for  centered at x = 8.  Determine T(x).  Also, compute T(7.5) rounded to the seventh decimal place, and compare T(7.5) with  rounded to the seventh decimal place.

 

 

 

13.  (4 points)  Determine the MacLaurin series for .  Write it in sigma notation.

 

 

 

14.  (4 points)  Does  diverge, converge conditionally, or converge absolutely?  Explain.