Page 95 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 8.9 Review Questions

1. Prove that a sequence cannot have more than one limit. (We, thus, speak of the limit of a
sequence.)
2. Give an example of a sequence that does not have a limit, or explain carefully why there
is no such sequence.
3. Give an example of a bounded sequence that does not have a limit, or explain carefully
why there is no such sequence.

4. Give a sequence (f ) of functions continuous on a set D with a limit that is not continuous.
n
5. Give a sequence of real functions differentiable on an interval which converges uniformly
to a non-differentiable function.


6. Find the set D of all z for which the sequence   n z n n  has a limit. Find the limit.
 z  3 

n



n

7. Prove that the series  a j  converges if and only if both the series  Rea j  and




 j 1   j 1 


 n 
 Ima j  converge.


 j 1 

n

1 
8. Explain how you know that the series    j  converges uniformly on the set |z|  5.
 
 j 1 z  
 
 
9. Suppose the sequence of real numbers () has a limit. Prove that
j
lim sum() = lim().
j j
10. For each of the following, find the set D of points at which the series converges:
n
j
(a)   j!z .  
  j 0  

n
j
(b)   jz .  
  j 0  

 n j 2 
(c)  j z . 
j
  j 0 3  

 n ( 1) j 

2 j
(d)  j 2 2 z  .
  j 0 2 (j!)  

11. Find the limit of
 n 
 (j 1)z .  
j


 j 0 

For what values of z does the series converge?
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