## Improper Integrals and Series: The Integral Test 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  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. [Geometry] [Algebra] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra] [Trigonometry] S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996