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Tuesday, 26 March 2013

LOCAL BEHAVIOR of FUNCTIONS PART 4


Consider a function f(x) which exhibits a Type I or Type II behavior on the interval [a,b] (in other words, the integral tex2html_wrap_inline128 is improper). We saw before that the this integral is defined as a limit. Therefore we have two cases:
1
the limit exists (and is a number), in this case we say that the improper integral is convergent;
2
the limit does not exist or it is infinite, then we say that the improper integral is divergent.
If the improper integral tex2html_wrap_inline128 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 tex2html_wrap_inline136 on [0,1]. We have
displaymath140
Therefore the improper integral
displaymath142
converges if and only if the improper integrals
displaymath144
are convergent. In other words, if one of these integrals is divergent, the integral tex2html_wrap_inline146 will be divergent.
The p-integrals Consider the function tex2html_wrap_inline148 (where p > 0) for tex2html_wrap_inline152 . Looking at this function closely we see that f(x) presents an improper behavior at 0 and tex2html_wrap_inline156 only. In order to discuss convergence or divergence of
displaymath158
we need to study the two improper integrals
displaymath160
We have
displaymath162
and
displaymath164
For both limits, we need to evaluate the indefinite integral
displaymath166
We have two cases:
if p=1, then we havedisplaymath170
if tex2html_wrap_inline172 , then we havedisplaymath174
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.

tex2html_wrap_inline182
If p <1, then we havedisplaymath186
and
displaymath188

tex2html_wrap_inline182
If p=1, then we havedisplaymath194
and
displaymath196

tex2html_wrap_inline182
If p > 1, we havedisplaymath202
and
displaymath204
The p-Test: Regardless of the value of the number p, the improper integral
displaymath158
is always divergent. Moreover, we have
tex2html_wrap_inline182
tex2html_wrap_inline212 is convergent if and only if p <1
tex2html_wrap_inline182
tex2html_wrap_inline218 is convergent if and only if p >1
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