Complete rewrite of Calculus/Series…
Note that "@@" marks things needing more attention before I post this in the main namespace.
- 1 Infinite series
- 2 Summation notation
- 3 Sequence of partial sums
- 4 Sum of an infinite series
- 5 Series convergence and divergence
- 6 Special forms
- 7 Properties of infinite series
- 8 Tests for convergence and divergence
- 8.1 Divergence test
- 8.2 Integral test
- 8.3 Comparison tests
- 8.4 Alternating series test
- 8.5 Root test
- 8.6 Ratio test
- 9 Strategy for testing infinite series
A series is the sum of a sequence of terms. An infinite series is the sum of an infinite number of terms (the actual sum of the series need not be infinite, as we will see below).
While there is usually some pattern in the terms being added, technically a series can be the sum of any sequence of terms:
The problem with the above series is, it is not at all clear what the next term of the series would be, nor any subsequent ones. To say anything useful about the series, there needs to be a clear pattern. Two such patterns seen in sequences that also appear in the study of series are called arithmetic and geometric.
An arithmetic series is the sum of a sequence of terms having a common difference (i.e., the difference between consecutive terms is always the same). For example,
is an arithmetic series with common difference 3, since , , and so forth. Unlike the first series above, this series has a clear pattern. It is obvious that the next term of the series is 16, and then 19, and so on.
A geometric series is the sum of a sequence of terms having a common ratio (i.e., the ratio between consecutive terms is always the same). For example,
is a geometric series with common ratio , since , , and so forth. As before, there is a definite pattern, and the next term of the series is clearly .
Summation notation provides a compact way of writing an infinite series. The arithmetic series above can be written
or, more simply
- "The sum, as goes from 1 to infinity, of..."
The algebraic expression on the right is the nth term of the series, and is often generically called "", as is also done with sequences. This means that the sum could have been written a bit more verbosely (and perhaps more pedantically) as
a form that more explicitly reflects the idea that a series is the sum of a sequence of terms. (The expression by itself defines the sequence of terms, but note that it is a bit ambiguous since it's not clear without additional context whether the sequence starts at or , or indeed any other value. Incidentally, the first summation formula above, which might have seemed a bit strange at first, came from the formula for the nth term of an arithmetic sequence, which you probably first learned in algebra class: .)
To verify that this summation formula represents the same sum as in the previous section, we simply evaluate the algebraic expression () for the first value of (in this case, 1), then the next value (2), and so forth:
As for the geometric series, it should be easy to verify the representation
Non-uniqueness of summation notation
Note that the representation of a series in summation notation is not unique, even ignoring simple algebraic rewrites. All of the sums seen so far have started at , as is usually the convention adopted for sequences, but we could have chosen to start the sums at any other value of . In fact, it is quite common to start infinite series at . If we follow that convention, the arithmetic series would be written
and the geometric series would be
Again, these expressions should be easy to verify.
Furthermore, as with functions, the variable used in the sum (called the index variable) is completely arbitrary. Thus, the following sums also represent the same series (respectively) as those above:
As a result of this fact, any series can be rewritten by simply changing the starting value of the series and compensating for this change in the formula for the nth term. To be specific, consider the first series above in this section. Using the substitution , we get the series
But then the variable can simply be changed back to to get
Notice how this new series can be seen as resulting from "bumping up" the starting value of the original series by 2 and "bumping down" each in the formula by the same amount (that is, replacing each by and then simplifying). This kind of change to a series is sometimes necessary to verify a given series formula (say, when looking up the answer to an exercise in a textbook).
Series versus sequence
As mentioned above, there is an important difference between an infinite series, which is the sum of an infinite number of terms, and the sequence formed by those same terms.
At the risk of being repetitive, our arithmetic series is
but the corresponding sequence of terms is
It might seem silly at this point to make such a big deal out of this distinction, but it will be very important going forward not to confuse these two concepts.
Sequence of partial sums
One special kind of sequence that is very useful to consider when studying series is the sequence of partial sums, defined in the following way (assuming a starting value of 1 for the index variable) for any positive integer :
For our arithmetic series, the sequence of partial sums is:
Note how this sequence of partial sums is very different from the sequence of terms discussed in the last section!
Just as the series itself can be written more compactly in summation notation by finding the pattern in its terms, the sequence of partial sums for a series can (sometimes) also be written compactly as a function of .
It should be easy to check that the following expression accurately represents the (first few terms of the) sequence of partial sums shown above:
To see where this expression came from, we first need to review some properties of sums that you probably remember from arithmetic and algebra, but might not be familiar with in summation notation.
Some properties of finite sums
Constant factors can be factored out of (or multiplied into) finite sums:
Finite sums can be added and subtracted, as long as they cover the same range of values of the index variable:
Putting these two ideas together, one may show the following property for arbitrary linear combinations:
Furthermore, there are special formulas for sums of certain simple expressions:
The first formula is obvious, since it represents the sum of n copies of the number 1. You might recognize the second formula as arising in the study of the binomial theorem and combinations (in particular, it represents the number of possible combinations of n distinct objects taken 2 at a time). The third formula is usually encountered in intermediate algebra or precalculus classes when geometric series are first studied.
With these facts, one may derive the expression given above for the sequence of partial sums of our arithmetic sequence.
Similarly, one may derive the sequence of partial sums for the geometric series:
Finding a series from its partial sums
Using the fact that
we see that
This provides a way of "recovering" the original series from its sequence of partial sums.
For our arithmetic series:
Sum of an infinite series
Now we can finally formally define the sum of an infinite series as the limit of its sequence of partial sums:
If the limit converges to a real number, say , then the infinite series is said to converge to the sum , or to be convergent with sum . If the limit diverges (including the cases where the limit is infinity or negative infinity), then the series is said to diverge or to be divergent in the same way, and its sum is said to not exist (or to be infinity or negative infinity, as appropriate). Note that one does not describe the series itself as "not existing" when it diverges, only its sum.
Consider again the arithmetic and geometric series we have been discussing up to this point. It is obvious by simply looking at the original arithmetic series
that it does not have a finite sum. The terms being added are themselves getting larger and larger without bound, so the sum is getting larger and larger without bound.
The sequence of partial sums given above formalizes this idea. In particular, because
the arithmetic series diverges to infinity.
Now consider the original geometric series:
The terms here are getting smaller and smaller, and indeed are approaching zero. Since adding zero to something doesn't change its value, it seems reasonable to suspect that the sum might be a fixed, finite number. What does the sequence of partial sums reveal?
So the geometric series does, in fact, converge to the finite sum 2.
It seems obvious that any series whose terms "blow up" to infinity (like our arithmetic series) will diverge, but does every series whose terms shrink to zero (like our geometric series) converge to a finite sum? It turns out the answer to that question is no.
There is a very important series whose sequence of terms goes to zero and yet the series diverges because the sequence of partial sums diverges to infinity. It is called the harmonic series:
Obviously the terms of this series go to zero:
But what about the sequence of partial sums? For convenience sake, we consider not the nth partial sum but the th partial sum, then group the terms in a clever way, and find a lower bound for each group of terms:
There are n terms equal to in the final sum, thus
The limit of this final expression, as , is infinity. This means the sequence of partial sums diverges. (Technically, we have only shown that a "subsequence" of the original sequence of partial sums diverges, but it turns out that this is sufficient to prove that the original sequence diverges.)
Therefore, the harmonic series diverges even though its terms shrink to zero. Many other series share this property, some of which will be discussed below.
When considering whether a given series is convergent or divergent, it is often useful to "ignore" the first several terms of the series. For example, as we have seen, the series
is geometric and converges to 2, but the series
is not geometric, since the first two terms don't fit the pattern of the rest of the series. But since the series behaves like a geometric series from the third term onward, one might call the series "eventually" geometric: it didn't start out that way, but eventually it settled down and behaved like a geometric series. We will use this idea of a series "eventually" having a certain property many times in the discussion that follows.
Clearly if the first (geometric) series above converges, the second (eventually geometric) series will also converge. In fact, by considering separately the sum of the first two terms and the sum of the rest of the terms, we can deduce that the sum of the second series is
Similarly, the series
converges in the same way as the original geometric series, but again the new (this time shorter) series converges to a different sum (the original sum minus the first two terms):
So, note that whether a series converges doesn't depend on what's happening at the beginning of the series, but the sum of the series definitely does!
Series convergence and divergence
Now that we've covered enough background about series, we can start to systematically investigate the convergence and divergence of many different kinds of infinite series. In some cases we will be able to find a formula for the partial sums, and thus the sum of the series, but in most cases we will not.
Necessary condition for convergence
Note that we have already considered one convergent series whose terms shrink to zero (the geometric series), one divergent series whose terms shrink to zero (the harmonic series), and one divergent series whose terms do not shrink to zero (the arithmetic series). What about the remaining case: a convergent series whose terms do not shrink to zero? Turns out, that's not possible.
Theorem: A necessary condition for series convergence
So, every convergent series has a sequence of terms that converges to zero. But, as we saw with the harmonic series, just because a series has terms that shrink to zero, that doesn't mean that the series converges. Therefore, having terms that go to zero is a necessary condition that a convergent series must satisfy, but not a sufficient condition that allows us to conclude that a given series converges. (Put more bluntly: Just because the terms go to zero doesn't mean the series converges!)
So what is a sufficient condition for series convergence? Well, there are many such conditions, several of which we will discuss below. All of them, however, will be saying essentially the same thing: that the terms of the series are going to zero "fast enough" that it allows the series to converge.
Sufficient condition for divergence
Recall from basic logic that the statement "If A then B" is equivalent to "If not B then not A". Applying this idea to the previous theorem gives a way of determining that an infinite series diverges (the so-called divergence test, which we will see again below).
Theorem: A sufficient condition for series divergence
While this gives a very useful way of checking for divergence, it is not a very "powerful" method, since — as we've already said — there are many divergent infinite series whose terms do actually go to zero. Several more powerful methods for checking divergence will be given below.@@
- to come@@
It is useful to distinguish between series with special forms, which all have specific formulas associated with them, and general-purpose tests that can be applied to a wide variety of different series.
Because the only arithmetic series that converges is the "trivial" series consisting of all zeros, we will not consider that type of infinite series any further.
A geometric series any series that can be written in either of the two equivalent forms
where is any nonzero real number and is any real number.
These two forms are the most common ones given in calculus textbooks, but note that both are special cases of the more general form
where, again, is any nonzero real number, any real number, and any integer.
Using any of the forms above, the constant is the first term of the series and the constant is the common ratio between successive terms; in particular,
for any for which both terms are defined.
The partial sums of such a geometric series are given by
The series converges if and only if , and in this case the sum of the series is
Note that this last formula can be used with any , but it only is meaningful (and represents the sum of the series) when , because otherwise the series doesn't even converge. In other words, be sure to check that the series converges before plugging into the formula for the sum.
As discussed in a previous section, the convergence or divergence of a series doesn't depend on where or how the series starts, so we could have defined a geometric series as starting at any value. But the formulas for the nth partial sum and the sum of the series given above only work when the first term is and the common ratio is . Consider the series
This is definitely a geometric series, but note that the first term is not , it is , so the partial sums and the sum of the series will not be as stated above. In this case, a simple algebraic rewrite fixes the problem:
We see, therefore, that the partial sums and sum of the series are the ones given above multiplied by .
To avoid having to rewrite an "obviously geometric" series just to get the same form as in the definition, the following general formulas can be used instead:
Again, these formulas should only be used after one has verified that the series is actually geometric and, if one is looking for the sum, that the series actually converges.
It's pretty obvious that this series is geometric with common ratio (because when increases by 1, the numerator increases by a factor of 3 and the denominator by a factor of 5) and that the first term is .
Since , the series converges to
To write the series in one of the definitional forms given above, notice that
which implies the same first term and common ratio as previously stated.
A telescoping series (or telescoping sum) is one that "expands" in such a way that most of its terms cancel away. Typically this will be a series of the form
where , , and are all integers with (assuming no division by zero occurs for any of the terms) and any nonzero real number.
The partial sums of such a series can be written in the form
by way of a partial fractions decomposition.
By expanding this finite series, one will see that everything cancels except portions of its first terms (i.e., the terms corresponding to ) and last terms (for ). It should then be easy to find the limit of the partial sums and hence the sum of the series.
As we will see later,@@ any telescoping series of the particular form shown above must converge, but there are telescoping series of other forms which do not converge.
This can be written as
and so is telescoping with , , , and .
The partial sums of the series are
by a partial fractions decomposition. Expanding the sum, we get
Notice that the fractions , , and immediately cancel, as do , , and . But by recognizing the pattern of cancelations, it should be clear that the fractions and also cancel at the beginning of the series (with later terms) and and also cancel at the end of the series (with earlier terms). Similarly, all the fractions in the omitted portion of the series () also cancel away. However, the fractions and at the beginning as well as and at the end do not cancel out. This is as we expect, since the fact that tells us that portions of the first two and last two terms will remain after canceling.
In this way we see that
and so the sum of the series is
A @@p-series is any series that can be written in the form
for some positive integer , nonzero real number , and real number . Such a series converges if and only if .
The partial sums of p-series are too complicated to write a general formula for, so we won't try to do this. Nor will we consider the sum that such a series converges to. There are methods that can be used to find the sums of certain particular p-series, but this will have to wait until we discuss power series.
Note that the harmonic series discussed above is a p-series with (and ).
Consider the series
This series can be written as
and so is a p-series with (and ). Since , the series diverges.
An alternating series is any series whose terms alternate in sign — that is, any series for which the product of any two consecutive terms is negative.
Equivalently, an alternating series is one that can be written in the form
for some fixed integers and , and sequence of positive terms . (If the nth term of the series is called , as usual, then notice that .)
As with geometric series, we have defined alternating series here in a slightly more general way than is typically done in calculus textbooks. Usually alternating series are either defined quite restrictively as
or as being in one of the two forms
in all cases the being some sequence of positive terms. It is easy to see that these other definitions are special cases of our formula above.
It should be obvious that we could not hope to write a formula for the partial sums of a general alternating series (besides, of course, the definition of partial sums given earlier), but, perhaps surprisingly, we can say when such a series converges.
An alternating series converges if its terms eventually decrease in magnitude to zero — that is, if
- the sequence is eventually decreasing.
However, if either of these conditions are not satisfied, it does not mean that the alternating series must diverge.
Note that any geometric series with is alternating. If , then the conditions for convergence of an alternating series will be satisfied. This can be proven in the general case,@@ but we will simply illustrate with an example.
Consider the series
This series can be written in the form
and so matches our definition of an alternating series (, , and ).
It is obvious that
is a decreasing sequence, so the series converges (as we knew it must, since it is geometric with ).
Consider the series
This is an alternating version of the harmonic series. Since
is a decreasing sequence of terms, the series converges.
Absolute and conditional convergence
At this point we have considered divergent series whose terms have no limit (the arithmetic series) and divergent series whose terms have a limit of zero (the harmonic series). But every convergent series must have terms that converge to zero. So, does this mean convergent series are "all the same"? Definitely not. There are two kinds of convergence that can be thought of as two "strengths" of convergence: absolute and conditional. The distinction is important because there are things you can do with absolutely convergent series that you cannot do with merely conditionally convergent ones. First, though, some definitions:
- A series is said to converge absolutely (or to be absolutely convergent) if converges.
- A series is said to converge conditionally (or to be conditionally convergent) if diverges but itself converges.
It should be obvious that this distinction only makes sense for series with a mixture of positive and negative terms. This includes, but is not limited to, alternating series.
We have seen that the harmonic series
is divergent but its alternating version
is convergent. Since the first series may be formed by taking the absolute value of the terms in the second series, we see that the second series is conditionally convergent.
Consider the series
This is an alternating series whose terms decrease to zero in magnitude, so it converges. Furthermore, the series formed by the absolute value of the terms
is a p-series with , so it converges also. Therefore the original alternating series is absolutely convergent.
Properties of infinite series
There are several properties of infinite series that are direct consequences of the corresponding properties of finite sums:
Constants can be factored out of (or multiplied into) infinite series:
- For any real number , converges if converges, in which case
- For any non-zero real number , diverges if diverges.
Note that converges if diverges but (because it is a series whose terms are all zero).
Sums and differences of convergent infinite series are convergent:
- If and both converge, then also converges and
- If and both converge, then also converges and
Sums and differences of one convergent and one divergent series are divergent:
- If converges and diverges, then and both diverge.
- If diverges and converges, then and both diverge.
Sums and differences of two divergent series may be convergent or divergent (thus one cannot decide whether they converge or diverge without applying a specific convergence or divergence test):
- If and both diverge, then and may either (both) converge or (both) diverge.
More generally, linear combinations of convergent infinite series are convergent:
- If and both converge and and are any real numbers, then also converges and
Linear combinations of a mix of convergent and divergent series are (usually) divergent:
- If converges and diverges, and and are any real numbers with , then diverges. (If , then converges.)
However, arbitrary linear combinations of divergent series are (usually) inconclusive with respect to their convergence without further testing:
- If and both diverge and and are non-zero real numbers, then may either converge or diverge. (If exactly one of the constants and is zero, then the latter series diverges. If both and , then the latter series converges.)
Special properties of absolutely convergent series
- to come@@
Tests for convergence and divergence
In this section we state and illustrate several general-purpose tests for convergence and divergence of infinite series. These tests, along with the special forms listed above are sufficient to classify most infinite series that come up in calculus classes as convergent (absolutely or conditionally) or divergent. However, not every test will work on a given series (that's why there are so many of them), so understanding which test is appropriate for which kind of series is crucial.@@
We have already seen the first general-purpose test for infinite series in the section above about a sufficient condition for divergence. It never concludes convergence, only divergence, and is therefore called the Divergence Test (it's also known as the nth-Term Test, or sometimes the Limit Test).
Theorem: Divergence test for infinite series
It is very important to remember that this test can only conclude divergence (hence the name); it cannot be used to conclude convergence. In particular, this means that if the limit of the terms is zero, then you don't conclude that the series is "not divergent" (since that means it is convergent). Instead, you say that the test is inconclusive and don't conclude anything about the series: some other test is then required to say whether the series converges or diverges.
Proof of p-series convergence criterion
Approximating the sum of a series with positive terms
Direct comparison test
Limit comparison test
Picking a series for comparison
Alternating series test
Absolute and conditional convergence
Approximating the sum of an alternating series