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
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
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
The Integral test
- The improper integral is convergent if and only if series is convergent
Example. Establish convergence or divergence of
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
- and .
are all divergent while the improper integrals
are all convergent.