Math 182

Feb 15, 2013

 

Determine whether or not each of the following improper integrals converges.

                                                                                

(diverges)                                                                                (converges)

 

 

                                                                                                                                         

 

 

 

 

 

                                                                                                           


Recall the comparison property for definite integrals:  If f(x) g(x) for all x in [a,b] then .

This property is very useful in determining whether some improper integrals converge too:

The comparison test:

If  0 ≤ f(x) ≤ g(x) for all x a,  and if   converges, then so does  .

If  f(x) ≥ g(x) ≥ 0 for all x a,  and if   diverges, then so does  .

 

 

 

 

 

 

 

 

 

 

 

 

 

Use a well chosen comparison to determine whether each of the following integrals converges or diverges:

                                                                             

 

Compare to                                                               compare to

 

 

 

 

 

 

 

Explain why a comparison with  will not help you to determine whether  converges or not.

 

(hint:  which is larger?  Why is that a problem?)

 

 

 

 

 

 

Does   converge?  Justify your answer!

 

(hint:  it does, but even though the previous comparison doesn’t help, another well chosen one might)