Consider a function f(x) which exhibits a Type I or Type II behavior on the interval [a,b] (in other words, the integral is improper). We saw before that the this integral is defined as a limit. Therefore we have two cases:
If the improper integral is split into
a sum of improper integrals (because f(x) presents more than one
improper behavior on [a,b]), then the integral converges if and only
if any single improper integral is convergent.
Example. Consider the function on [0,1]. We have
Therefore the improper integral
converges if and only if the improper integrals
are convergent. In other words, if one of these integrals is
divergent, the integral
will be divergent.
The p-integrals Consider the function (where p > 0) for . Looking at this function closely we see that f(x) presents an improper behavior at 0 and only. In order to discuss convergence or divergence of
we need to study the two improper integrals
We have
and
For both limits, we need to evaluate the indefinite integral
We have two cases:
In order to decide on convergence or divergence of the above two improper integrals, we need to consider the cases: p<1, p=1 and p >1.
and
and
and
The p-Test: Regardless of the value of the number p, the improper integral
is always divergent. Moreover, we have
In the next pages, we will see how some easy tests will help in deciding whether an improper integral is convergent or divergent.
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Mohamed A. Khamsi