LOCAL BEHAVIOR of FUNCTIONS PART 5

Improper integrals and series have a lot in common. The integral test bridges the two notions. Notice that series do possess tools which are not available for improper integrals (such as the ratio and root tests) and the improper integrals possess other tools not available for series (such as the techniques of integration). So depending on the nature of the problem, you may switch from one to the other one via the integral test.
The Integral Test. Consider a decreasing function  . Hence for any  , we have

which implies

Set  , then we have

for  . If we add these inequalities from n=1 to n=N, we get

If  is the sequence of partial sums associated to the series  , then we have

or equivalently


Since  for any  , then we know that

the series  is convergent if and only if the sequence  is bounded;

the improper integral  is convergent if and only if the sequence
is bounded.

Using the above inequalities, we conclude:
The Integral test

The improper integral  is convergent if and only if series  is convergent

Remark. Note that it may happen that f(x) is not decreasing on the entire interval  but only on some subinterval  (where A > 1). The above conclusion is still valid.
Example. Establish convergence or divergence of

Answer. Set

Then we have

Clearly the function  for  . Hence f(x) is decreasing on  . So the integral test implies that the improper integral

is convergent if and only if the series

is convergent. We recognize Bertrand's series. So we conclude that the improper integral

is convergent if and only if

 or

 and  .

So for example, the improper integrals
are all divergent while the improper integrals

are all convergent.