Friday, April 11, 2008


Practice Test 3

The real test is Monday, April  14.

1.  (a)  Write the power series for each of these.

    (b)  Give the radius of convergence.

    (c)  Give an error bound for truncating the series at degree 7 

          on the interval  ( -1/2, 1/2 ).  

         1
(a)   --------
        1-x

          
(b) ln(1+x)


c) ArcTan[x]
            
       x
(d)   e


(e) Sin[x]


(f) Cos[x]


(g) Sinh[x]


(h) Cosh[x]


2.  (a)  Write the power series for  y = Cos[Sqrt[x] ]. 

    (b)  What is its radius of convergence.

    (c)  What is the error by truncating the series at degree 8 on

the interval [-8,8];

                                              2
3.  (a)  Write the power series for y = ln(1+x  ).

    (b)  What is its radius of convergence.

    (c)  How what degree polynomial is necessary to have an accuracy of 0.00001

         on the interval (-1/2,1/2) ?
  

                                                                   2
4.  (a)  Write the degree 3 MacLaurin polynomial for  y =  ln(1+x+x  ) 


    (b)  Give a bound on the accuracy of your approximation on (-1/2,1/2).







5.  Find the radius of convergence of

                   n
      Infinity    3      n
        SUM     ------  x
        n=1         n
                 1+2

6.  Prove that the series converges

     Infinity    Sin[n]
        SUM    -------- 
        n=1        n
                  3
     
7.  Prove that the series converges
  
       Infinity   Sin[1/n]         
         SUM     ----------
         n=1         2
                    n

8.  Prove that the series converges

        Infinity       2n+1  
          SUM        --------
          n=1        2
                    n  + n + 1




                         th
9.  State and prove the n   term test for divergence.


                          th
10.  State and prove the n   term test for convergence of

     an alternating series.



11.  State and prove the integral test.



12.  State the ratio test.