Calculus 2  Hour Exam 2   -   Math 231     Spring, 2012

 

 

 

 

 

1.

 

 

Determine if the following sequences converge or diverge.  If the sequence converges,

then find the value that it converges to.

 

a.        {an} = n8/en                             

 

b.      {an} =  (3n e2/n / (5n + 4)

 

c.       {an} = eln  + (-1)n / √ (n2 + 1 )

 

Answers:  a. Converges to 0,   b.  Converges to 3/5    c.  Converges to 1

 

 

 

 

 

 

2.

 

Determine if the series converge or diverge.  If it converges find the value it

converges to.

             ∞

a.       ∑   (n – 1) / (9n + 8)

     n = 1

 

            ∞

b.       ∑   [ 2n + (-1)n ] / 3n      Answers:  a. diverges    b.  Converges to 2.25

     n = 1

 

 

 

 

 

3.

 

 

Determine if the following statements are true or false.

                                                                   ∞                     ∞

a.       Whenever  0 ≤  an  ≤  100 bn  and  Σ an diverges,  Σ bn also diverges.      

                                                                   n=1                 n=1

                                                         ∞

b.      Whenever  lim an = 0, then  Σ an converges.

                 n→∞                  n=1

 

Answer:  a.  True      b.  False

 

 

 

 

 

 

 

 

4.

 

Give precise and concise definitions.  If the theorems have hypotheses you must clearly

state the hypotheses to receive credit.

                                                                               ∞

a.          Define the partial sum  sn  of the series  Σ an   Answer  sn  = 

                                                                             n=1

b.         State the Integral test for series.

 

c.          State the Montone Convergence Theorem for sequences.

 

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