Conditional Convergence

We have seen that, in general, for a given series tex2html_wrap_inline335 , the series tex2html_wrap_inline337 may not be convergent. In other words, the series tex2html_wrap_inline335 is not absolutely convergent. So, in this case, it is almost a lost case, meaning it is very hard to use the old tools developed for positive series. But, for a very special kind of series we do have a partial answer (due to Abel). This is the case for alternating series. These are of the form

displaymath341,

where tex2html_wrap_inline343 .

Example: The series

displaymath345

is alternating. Note that we do not have tex2html_wrap_inline347, but tex2html_wrap_inline349 . This is not important. What matters is that the sign alternates; once it is positive, the next one is negative and so on....

Main Problem: When does an alternating series converge?

In order to appreciate Abel's result on alternating series, let us play with the above example. Indeed, consider the series

displaymath351.

Let us generate the sequence of partial sums tex2html_wrap_inline353 . We have

displaymath355

This clearly implies

displaymath357.

So, one may wonder whether the even sequence tex2html_wrap_inline359 is increasing and the odd sequence tex2html_wrap_inline361 is decreasing while satisfying

displaymath363.

The answer is: YES. Indeed, we have

displaymath365,

which implies tex2html_wrap_inline367 . Let us check that tex2html_wrap_inline359 is increasing (the odd one is left to the reader to prove). We have

displaymath371.

Since

displaymath373,

then we have tex2html_wrap_inline375, which implies that tex2html_wrap_inline359 is increasing. By the way, we did evaluate tex2html_wrap_inline379, but, in fact, we did not need that. What makes it work is the fact that the sequence tex2html_wrap_inline381 is decreasing. So, since the sequence tex2html_wrap_inline359 is increasing and bounded above by tex2html_wrap_inline385 , then it is convergent to a number A. The same holds for tex2html_wrap_inline361 which is decreasing and bounded below by tex2html_wrap_inline389 . Hence, it is convergent to a number B. We must have

displaymath391.

This obviously implies

displaymath393.

Since

displaymath395,

we deduce that we must have A = B, which gives

displaymath399.

This clearly implies that the sequence tex2html_wrap_inline353 is convergent and

displaymath403.

Therefore, the series

displaymath351

is convergent and we have

displaymath407.

From the above inequalities, we get

displaymath409,

for any tex2html_wrap_inline411 . These inequalities allow for an approximation of the total sum by the partial sums. If you wonder what the total sum is, the answer is (by using Taylor series):

displaymath413.

Remark: Let us give another way to prove

displaymath413.

First, consider the sequence tex2html_wrap_inline417 defined by

displaymath419.

It is easy to check that tex2html_wrap_inline421, for tex2html_wrap_inline411 . Let us show that tex2html_wrap_inline417 is decreasing. Indeed, we have

displaymath427.

Set

displaymath429.

We have

displaymath431.

Hence, the function f(x) is increasing for tex2html_wrap_inline435 . Since

displaymath437,

then we must have tex2html_wrap_inline439 for tex2html_wrap_inline441 . This implies

ln$\displaystyle \left(\vphantom{1 + \frac{1}{x} }\right.$1 + $\displaystyle {\frac{{1}}{{x}}}$$\displaystyle \left.\vphantom{1 + \frac{1}{x} }\right)$ $\displaystyle \geq$ $\displaystyle {\frac{{1}}{{x+1}}}$

for tex2html_wrap_inline441 . Hence, we have

ln$\displaystyle \left(\vphantom{\frac{n+1}{n} }\right.$$\displaystyle {\frac{{n+1}}{{n}}}$$\displaystyle \left.\vphantom{\frac{n+1}{n} }\right)$ $\displaystyle \geq$ $\displaystyle {\frac{{1}}{{n+1}}}$

for tex2html_wrap_inline411 . Therefore, the sequence tex2html_wrap_inline417 is decreasing. Since it is bounded below by 0, we conclude that tex2html_wrap_inline417 is convergent. Write

displaymath455.

C is called Euler's constant. We have

displaymath459,

for any tex2html_wrap_inline411 . Let us go back to the alternating series

displaymath351.

We have

displaymath465.

Since

displaymath467,

we conclude that

displaymath469.

So what did we learn from the above example? By looking carefully at the above calculations, we may be able to come up with a more general result.

Alternating Series Test:

Consider the alternating series

displaymath471

where tex2html_wrap_inline473 . Assume that:

1.
tex2html_wrap_inline475 is decreasing;
2.
tex2html_wrap_inline477;
Then the series

displaymath471

is convergent. Moreover, the estimate of the total sum

displaymath481,

by the nth partial sum tex2html_wrap_inline483, has error of magnitude tex2html_wrap_inline485 at most. In other words, we have

displaymath487.

Example: Classify the series

displaymath489

as either absolutely convergent, conditionally convergent, or divergent.

Answer: Consider the series of the absolute values

displaymath491.

This is a Bertrand Series with tex2html_wrap_inline493 and tex2html_wrap_inline495 . Using the Bertrand Series Test, we conclude that it is divergent. Hence, the series

displaymath489

is not absolutely convergent. Since this series is alternating, with

displaymath499,

let us check if the assumptions of the Alternating Series Test are satisfied. First, we need to check that tex2html_wrap_inline501 is decreasing. Set

displaymath503.

We have

displaymath505.

Clearly, we have f '(x) < 0, for x > e. Hence, the sequence tex2html_wrap_inline511 is decreasing. It is easy to check that

displaymath513.

Therefore, all the Alternating Series Test assumptions are satisfied. We then conclude that the series

displaymath489

is convergent. In fact, in order to be precise it is conditionally convergent.

Example: Classify the series

displaymath517

as either absolutely convergent, conditionally convergent, or divergent.

Answer: It is not clear from the definition what this series is. So we advise you to take your calculator and compute the first terms to check that in fact we have

displaymath519

This is the case because

displaymath521

So, this is an alternating series with tex2html_wrap_inline523 . Since this sequence is decreasing and goes to 0 as tex2html_wrap_inline525 , then by the Alternating Series Test, the series

displaymath527

is convergent. Note that it is not absolutely convergent.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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