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Tuesday, 26 March 2013

The Particular Case of Positive Series


Consider the series tex2html_wrap_inline264 and its associated sequence of partial sums tex2html_wrap_inline266 . Here we will assume that the numbers we are about to add are positive, that is, tex2html_wrap_inline268 for any tex2html_wrap_inline270 . It is clear that the process of generating the partial sums will lead to an increasing sequence, that is,
displaymath272,
for any tex2html_wrap_inline270 . Our previous knowledge about increasing sequences implies the following fundamental result:

The positive series tex2html_wrap_inline276 is convergent, if and only if, the sequence of partial sums tex2html_wrap_inline266 is bounded; that is, there exists a number M > 0 such thatdisplaymath282,
for any tex2html_wrap_inline270 .
This result has many implications. For example, we have the following result (called The Basic Comparison Test):
Consider the positive series tex2html_wrap_inline286 .
1.
Assume there exists a convergent series tex2html_wrap_inline288 such thatdisplaymath290,
then the series tex2html_wrap_inline264 is convergent.
2.
Assume there exists a divergent series tex2html_wrap_inline294 such thatdisplaymath296,
then the series tex2html_wrap_inline264 is divergent.
Recall that in previous pages, we showed the following
displaymath300,
which means that the series tex2html_wrap_inline302 is divergent.
Example: Show that the series
displaymath304
is divergent.
Answer: We have tex2html_wrap_inline306, for any tex2html_wrap_inline270 ; hence
displaymath310.
Since tex2html_wrap_inline302 is divergent, we deduce from the Basic Comparison Test, that tex2html_wrap_inline314 is divergent.
Example: Show that the series
displaymath316
is convergent.
Answer: Since
displaymath318
and the geometric series tex2html_wrap_inline320 is convergent, then the series tex2html_wrap_inline322 is convergent (using the Basic Comparison Test).
The next result (known as The p-Test) is as fundamental as the previous ones. Usually we combine it with the previous ones or new ones to get the desired conclusion.
Consider the positive series (called the p-seriestex2html_wrap_inline324 . Since the limit of the numbers must add to 0, in order to expect convergence, we assume that p > 0. The next result deals with convergence or divergence of the series
displaymath328
when p >0.
The p-series tex2html_wrap_inline332 converges, if and only if, tex2html_wrap_inline334 .
The proof of the above result is very instructive by itself. So let us discuss how this works:

Consider the function tex2html_wrap_inline336 defined bydisplaymath338.
It is easy to check that f(x) is decreasing on tex2html_wrap_inline342 . Hence, for any tex2html_wrap_inline270 , we have for any tex2html_wrap_inline346 ,
displaymath348
which implies
displaymath350,
that is,
displaymath352.
Using this inequality, we get
displaymath354
and
displaymath356.
If tex2html_wrap_inline358 , then we have
displaymath360
and if p=1, then we have
displaymath364.

We have three cases:
Case 1: p < 1, then we havedisplaymath368.
Since tex2html_wrap_inline370 , then the series tex2html_wrap_inline324 is not bounded,and therefore it is divergent.
Case 2: p > 1, then we havedisplaymath376
but, since
displaymath378,
we get
displaymath380,
which means that the sequence of partial sums associated to the series tex2html_wrap_inline324 is bounded. Therefore, the series tex2html_wrap_inline324 is convergent.
Case 3: p = 1, we have already shown that the series tex2html_wrap_inline302 is divergent. Though one may want to easily check that we havedisplaymath390,
which shows that the sequence of partial sums is not bounded.
Example: Discuss the convergence or divergence of
displaymath392.
Answer: It is not hard to show that for any tex2html_wrap_inline270 , we have tex2html_wrap_inline396 . Then, we have
displaymath398.
Since, by the p-Test, the series tex2html_wrap_inline400 is convergent, the Basic comparison Test implies that tex2html_wrap_inline402 is convergent.
The last result on positive series may be the most useful of all. Indeed, the Limit Test should be always in mind when it comes to cleaning up some undesirable terms.
Before we state this test, we need a new notation. Indeed, we will say that the two sequences tex2html_wrap_inline404 and tex2html_wrap_inline406 are equivalent, or tex2html_wrap_inline408, if and only if,
displaymath410.


Let tex2html_wrap_inline276 and tex2html_wrap_inline414 be two positive series such thatdisplaymath416.
Then tex2html_wrap_inline276 converges, if and only if, tex2html_wrap_inline414 converges.
Example: Determine whether the series
displaymath422
is convergent or not.
Answer: Note that when n is large we have tex2html_wrap_inline426 and tex2html_wrap_inline428 . Then it is easy to check that
displaymath430.
Using the p-test we get that the series tex2html_wrap_inline432 is convergent. Hence, by the Limit-test, we deduce the convergence of the series tex2html_wrap_inline434 .
Example: Determine whether the series
displaymath436
is convergent or not.
Answer: We know that
displaymath438.
This limit means that when x is very small tex2html_wrap_inline442 , then tex2html_wrap_inline444 (a very useful conclusion in physics, for example, when dealing the motion of the pendulum). So when n is large, 1/n will be small and thereforetex2html_wrap_inline450 . This clearly implies that
displaymath452
Since the series tex2html_wrap_inline302 is divergent, the limit-test implies that the series tex2html_wrap_inline456 is divergent.
Remark: It should be appreciated that, without the Limit-test, it would be very hard to check the convergence.
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