Calculus 2  Hour Exam 2   -   Math 231     Spring, 2010

 

 

 

 

 

1.

 

 

For which real number  α  does the series

             ∞

             ∑ (-1)n - 1 nα        Hint:  Use alternating series test and p-series

     n = 1

 

converge conditionally?  For which  α  does it converge absolutely?

 

Answers:  Conditional convergence for  -1 ≤ α < 0   Absolute convergence for  α  <  -1

 

 

 

 

2.

 

Determine if the series converges or diverges.  For convergent series find the sum.

 

             ∞

a.       ∑  [√ ln(n)] / n     Hint:  Try integral test         Answer:  Diverges

     n = 2

 

 

 

 

3.

 

 

Determine if the series converges or not.

 

        ∞

        ∑ (-1)n - 1  n 1/n      Hint:  Try nth term test for divergence     Answer:  Diverges

      n = 1

 

 

 

4.

 

 

Determine if the sequence   an  =  n[ cos(1/n) – 1 ] converges.  If it converges, then

find its limit.

 

 

Hint:  Try the substitution law for a sequence.       Answer:  Converges to zero.

 

 

 

 

5.

 

Determine if the series converges.

             ∞

         ∑  3n n! / nn           Hint:  Try ratio test.        Answer:  Diverges

            n = 1