Infinite Series and Other Limiting Processes
The geometric series, Taylor series and several special examples, which we have already met, suggest that we may well study from a rather more general point of view those limiting processes under the heading of summation of infinite series . By its nature, any limiting value
can be written as an infinite series, because, if n takes the values 1, 2, 3, ···, we need only set an = sn - sn-1 for n > 1 and a1 = s1 , in order to obtain
and the value S thus appears as the limit of sn - the sum of n terms as n increases. We express this fact by saying that S is the sum of the infinite series
.
Thus, an infinite series is simply a way of representing a limit, where each successive approximation is found from the preceding one by addition of one more term. In principle, the expression of a number as a decimal is merely the representation of a number a in the form of an infinite series a = a1+a2+a3+ ··· where, if 0 £ a £ 1, the term an is put equal to an´10-n and an is a whole number between 0 and 9, inclusively. Since every limiting value can be written in the form of an infinite series, it may seem that a special study of series is superfluous. However, in many cases, it happens that limiting values occur naturally in the form of infinite series, which often exhibit particularly simple laws of formation. Of course, it is not true that every series has an easily recognizable law of formation. For example, the number p can certainly be represented as a decimal, yet we know no simple law which enables us to state the value of an arbitrary digit, say its 7000-th decimal. However, if we set aside the representation of p by a decimal and consider instead Gregory's series, we have an expression with a perfectly clear general law of formation.
Infinite series, in which approximations to the limit are found by repeated addition of new terms, are analogous to infinite products, in which the approximations to the limit arise from repeated multiplication by new factors. However, we shall not go deeply into the theory of infinite products; the principal subject of this chapter and of Chapter IX will be infinite series.
8.1 The Concepts of Convergence and Divergence
8.1.1 The Fundamental Ideas: We consider an infinite series the general term of which we denote * by an; the series then has the form
The symbol with the summation sign on the right hand side is merely an abbreviated way of writing the expression on the left hand side.
* For formal reasons, we include the possibility that certain of the numbers an may be zero. If all an vanish onwards from a number N (i.e., when n > N), we speak of a terminating series.
If, as n increases, the n-th partial sum
approaches a limit
we say that the series converges, and
otherwise that it diverges. In the first case, we call S the sum
of the series. We have already
met with many examples of convergent series; for instance, the
geometric series 
which converges to the sum l/(l-q) for q
< 1, Gregory's series, the series for log 2, the series for e,
and others converge. In the language of infinite series, Cauchy's convergence test is
expressed as follows:
A necessary and sufficient condition for the convergence of a series is that the number
becomes arbitrarily small if m and n are chosen sufficiently large (m > n).
In other words: A series converges if and only if the following condition is met: Given a positive number e , however small, it is possible to choose an index N=N(e), which, in general, increases beyond all bounds as e ® 0, in such a way that the |sm - sn| is less than e , provided only that m > N and n > N.
We can make the meaning of the convergence test clearer by considering the geometric series with q = 1/2. If we choose e = 1/10, we need only take N = 4, because
and
If we choose e equal to 1/100, it is sufficient to take 7 as the corresponding value of N, as is easily verified.
Obviously, it is a necessary condition for the convergence of a series that
Otherwise, undoubtedly, the convergence criterion cannot be fulfilled. However, this necessary condition is by no means sufficient for convergence; on the contrary, it is easy to find infinite series the general term an of which approaches 0 as n increases, yet their sum does not exist, because the partial sum sn increases without limit as n increases.
An example of this is the series
with the general
term 
. We immediately see that
The n-th partial sum increases beyond all bounds as n increases, whence the the series diverges
The same is true for the classical example of the harmonic series
for which
Since n and m = 2n can be taken as large as we please, the series diverges, because Cauchy's test is not fulfilled; in fact, the n-th partial sum tends obviously to infinity, since all the terms are positive. On the other hand, the series, formed from the same numbers with alternating signs,
converges and has the sum log 2.
It is by no means true that in every divergent series sn tends to +¥ or -¥. Thus, in the case of the series
we see that the partial sum sn has alternately the values 1 and 0, and, on account of this backwards / forwards oscillation neither approaches a definite limit nor increases numerically beyond all bounds.
As the convergence and divergence of an infinite series is concerned, the following fact which, though self-evident, is very important should be noted: The convergence or divergence of a series is not changed by insertion or removal of finite numbers of terms. As far as convergence or divergence is concerned, it does not matter in the least whether we begin the series at the term a0, or a1, or a5, or at any other arbitrarily chosen term.
8.1.2 Absolute Convergence and
Conditional Convergence:. The series 
diverges;
but if we change the sign of every second term, the resulting
series converges. On the other hand, the geometric series 1-q+q²-q³···
converges and has the sum l/(l + q), provided that 0 £ q < 1; on
converting all the signs to +, we obtain the series
which also converges and has the sum l/(l - q).
There appears here a distinction which we must examine more closely. With a series all terms of which are positive, there are only two possible cases; either it converges or the partial sum increases beyond all bounds with n. In fact, the partial sums, being a monotonic increasing sequence, must converge, if they remain bounded. Convergence occurs, if the terms approach zero sufficiently rapidly as n increases; on the other hand, divergence occurs, if the terms do not approach zero at all or if they approach zero too slowly. However, in series, where some terms are positive and others negative, the changes of sign may bring about convergence, since a too great increase in the partial sums, due to the positive terms, is compensated by negative terms, as a final result of which a definite limit is approached.
In order to grasp this fact better with a
series 
, which has positive and negative terms, we compare
it with the series which has the same terms all with positive
signs, that is,
If this series converges, then for sufficiently large values of n and m > n, the expression
will certainly be as small as we please; on account of the relation
the expression on the left hand side is also
arbitrarily small, whence the original series converges. In this
case, the original series is said to be absolutely
convergent. Its convergence is
due to the numerical smallness of its terms and does not depend
on the change of the signs.
On the other hand, if the series with all the terms taken positively diverges and the original series still converges, we say that the original series is conditionally convergent. Conditional convergence results from the terms of opposite signs compensating one another.
For conditional convergence, Leibnitz's
convergence test is frequently
of use: If the terms of a
series have alternating sign and, moreover, their absolute values
|an| tend monotonically to 0 (so
that |an+1| < |an|),
the series converges. (Example: Gregory's
series).
In the proof, we assume that a1 > 0, which does not essentially limit the generality of the argument, and write our series in the form
where all the terms bn are now positive, bn tends to 0 and the condition bn+1<bn is satisfied. If we bracket the terms together in the two ways
we see at once that the partial sums satisfy the two relations:
On the other hand, s2n < s2n+1 < s1 and s2n+1 > s2n > s2, whence the odd partial sums s1, s3, ··· form a monotonic decreasing sequence, which in no case drops below the value s2 and this sequence has a limit L (c.f. A1.1.4). The even partial sums s2, s4, ··· likewise form a monotonic increasing sequence the terms of which in no case exceed the fixed number s1, whence this sequence must have a limiting value L'. Since the numbers s2n and s2n+1 differ only by the number b2n+1, which approaches 0 as n increases, the limiting values L and L' are equal to each other. In other words, the even and the odd partial sums
approach the same limit, which we will now denote by S (Fig. 1). However, this implies that our series is convergent, as has been asserted; its sum is S.
In conclusion, we make another
general remark about the fundamental difference between absolute convergence and conditional
convergence. We consider a convergent
series and denote the positive terms of the series by p1,
p2, p3, ··· and the
negative terms by -q1, -q2,
-q3, ···. If we form the n-th
partial sum 
of the given series, a certain nmnber, say, n'
positive terms and a certain number, say n"
negative terms must appear, where n'+n"=n.
Moreover, if the number of positive terms as well as that of
negative terms in the series is infinite, then the two numbers n'
and n" will increase beyond all bounds with n.
We see immediately that the partial sum sn
is simply equal to the partial sum 
of the positive terms
of the series plus the partial sum 
of the negative terms.
If the given series converges absolutely, then both the series of
positive terms 
and the series of the absolute values of the
negative terms 
certainly converge. In fact, as m
increases, the partial sums 
are monotonic non-decreasing with the
upper bound 
.
The sum of an absolutely convergent series is then simply equal to the sum of the series consisting only of the positive terms plus the sum of the series consisting of the negative terms, or, in other words, it is equal to the difference of the two series with positive terms.
However, 
; as n increases, n' and n"
must also increase beyond all bounds, and the limit of the left
hand side must therefore be equal to the difference of the two
sums on the right hand side. If the series contains only a finite
number of terms of one particular sign, the facts are
correspondingly simplified. On the other hand, if the series does
not converge absolutely, but does converge conditionally, then
both the series and must be divergent, because, if both were convergent,
the series would converge absolutely, contrary to our hypothesis.
If only one diverges, say , and the other converges, then separation
into positive and negative parts, 
shows that the series
could not converge; in fact, as n increases, n'
and 
would
increase beyond all bounds, while the term 
would approach a
definite limit, so that the partial sum sn
would increase beyond all bounds.
Hence, we see that a conditionally convergent series cannot be thought of as the difference of two convergent series, one consisting of its positive terms and the other consisting of the absolute values of its negative terms.
Closely connected with this fact, there arises another difference between absolutely and conditionally convergent series which we shall now mention briefly.
8.1.3 Rearrangement of Terms: It is a property of finite sums that we may change the order of the terms or, as we say, rearrange the terms at will without changing their values. There arises the question: What is the exact meaning of a change of the order of terms in an infinite series and does such a rearrangement leave the value of the sum unchanged? While in the case of finite sums there arises no difficulty, for example, in adding the terms in reverse order, in the case of infinite series such a possibility does not exist; there is no last term with which to begin. Now a change of the order of terms in an infinite series can only mean this: A series a1 + a2 + a3 + ··· is rearranged into a series b1 + b2 + b3 + ··· , provided that every term an of the first series occurs exactly once in the second series and conversely. For example, the amount by which an is displaced may increase beyond all bounds as n does; the only point is that it must appear somewhere in the new series. If some of the terms are moved to later positions in the series, other terms must, of course, be moved to earlier positions. For example, the series
is a rearrangement of the geometric series l + q + q² + ···.
As regards the change of order, there is a fundamental distinction between absolutely convergent and conditionally convergent series. In absolutely convergent series, rearrangement of terms does not affect the convergence and the value of the sum of the series is not changed, exactly as in the case of finite sums.
On the other hand, in conditionally convergent series, the value of the sum of the series can be changed at will by suitable rearrangement of the series, and the series can even be made to diverge, if so desired.
The first of these facts,
referring to absolutely convergent series, is easily established.
Let us assume, to begin with, that our series has only positive
terms and consider the n-th partial sum 
All the
terms of this partial sum occur in the m-th partial sum 
of the
rearranged series, provided only that m is chosen large
enough. Hence tm ³ sn.
On the other hand, we can determine an index n' so large
that the partial sum 
of the first series contains all the terms
b1, b2 , ··· , bm.
It then follows that tm£sn'£ A, where
A is the sum of the first series. Thus, for all
sufficiently large values of m, we have sn
£ tm £ A; since sn
can be made to differ from A by an arbitrarily small
amount, it follows that the rearranged series also converges, in
fact, to the same limit A as the original series.
If the absolutely convergent series has both positive and negative terms, we may regard it as the difference of two series each of which has only positive terms. Since in the rearrangement of the original series each of these two series merely undergoes rearrangement and therefore converges to the same value as before, the same is true of the original series when rearranged. In fact, by the case just considered, the new series is absolutely convergent and is therefore the difference of the two rearranged series of positive terms.
The fact just proved may seem to the beginner to be a triviality. The fact that it really does require a proof and that in this proof absolute convergence is essential can be demonstrated by an example of the opposite behaviour of conditionally convergent series. Consider the familiar series
Write below it the result of multiplication by the factor 1/2
and add the two series, combining the terms in vertical columns. We thus obtain
This last series can obviously be obtained by rearranging the original series and yet the value of the sum of the series has been multiplied by the factor 3/2. It is easy to imagine the effect of the discovery of this apparent paradox on the mathematicians of the Eighteenth Century, who were accustomed to operate with infinite series without regard to their convergence.
We shall give the proof of the theorem stated above concerning the change in the sum of a conditionally convergent series which arises from a change of order of its terms, although we shall have no occasion to make use of the result.
Let p1, p2,
··· be the positive terms and -q1, -q2,
··· the negative terms of the series. Since the absolute value
|an| tends to 0 as n
increases, the numbers pn and qn
must also tend to 0 with n. Moreover, as we have already
seen, the sum 
must diverge and the same is true for 
This abbreviated notation for and
analogous expressions for other series will be used often in what
follows.
We can now find easily a rearrangement of the
original series which has an arbitrary number a as its
limit. In order to be specific, let a be positive. We
then add the first n1 positive terms, just
enough to ensure that the sum 
is less than a.
Since the sum increases with n1 beyond all
bounds, it is always possible, by using enough terms, to make the
partial sum larger than a. The sum will then differ from
the exact value a by at most pn.
We now add just enough negative terms 
in order to ensure that
the sum 
is less than a. This is also possible, as
follows from the divergence of the series from the divergence of
the series . The difference between this sum and a is
now at most 
. We now add just enough other positive terms 
in order
to make the partial sum again larger than a, as is again
possible since the series of positive terms diverges. The
difference between the partial sum and a is now at most 
. We again
add just just enough negative terms 
, which start next after
the last one previously used, to make the sum once more less than
a and continue in the same way. The values of the sums
thus obtained will oscillate about the number a and,
when the process is carried out far enough, the oscillation will
only take place between arbitrarily narrow bounds; in fact, since
the terms pn and qn themselves tend to 0 when n is sufficiently
large, the length of the interval in which the oscillation occurs
will also tend to 0, and the theorem is proved.
In the same manner, we can rearrange s series in such a way as to make it diverge; we merely have to choose such large numbers of positive terms as compared with the negative ones that there occurs no longer compensation.
8.1.4 Operations with Infinite Series: It is clear that two convergent infinite series a1 + a2 + ··· = S and b1 + b2 + ··· = T can be added term by term, i.e., that the series formed from the terms cn = an + bn converges and has the sum S + T, because
This theorem is really nothing more than another statement of the fact (1.6.4) that the limit of the sum of two terms is the sum of their limits.
It is also clear that, if we multiply each term of a convergent infinite series by the same factor, the series remains convergent and its sum is multiplied by the same factor.
In the cases just mentioned, it is immaterial whether the convergence is absolute or conditional. On the other hand, a further study, which is not necessary for us here, shows that, if two infinite series are multiplied by each other by the method used in multiplying finite sums, the product series will not usually converge or have the product of the two sums as its sum unless at least one of the two series is absolutely convergent.
Exercises 8.1: Prove that
4. For what values of a does the series 
converge? 5.* Prove that, if 
converges and an
= a1 + a2 + ···
+an, then the sequence
also converges and has the limit 
6. Does the series 
converge?
7. Does the series 
converge?
8.2 Tests for Convergence and Divergence
We have already encountered a test of a general nature for the convergence of series, which applies to series with terms of alternating signs and decreasing absolute values and which asserts that such series are at least conditionally convergent. In what follows, we shall only consider criteria referring to absolute convergence.
8.2.1 The Comparison Test: All such considerations of convergence depend on the comparison of the series in question with a second series; this second series is chosen in such a way that its convergence can readily be tested. The general comparison test may be stated as follows:
If all the numbers b1,
b2 ··· are positive and the series 
converges
and if
for all values of n,
then the series 
is absolutely convergent.
If we apply Cauchy's test, the proof becomes almost trivial, because, if m ³ n, we have
Since the series converges, the right hand side
is arbitrarily small, provided that n and m are
sufficiently large, whence for such values of n and m
the left hand side is also arbitrarily small, so that, by
Cauchy's test, the given series converges. The convergence is
absolute, since our argument applies equally well to the
convergence of series of absolute values |an|.
The analogous proof for the following fact will be left to the reader:
If 
and the series diverges,
then the series is certainly not absolutely convergent.
8.2.2 Comparison with the Geometric Series: In applications of the test, the most frequently used comparison series is the geometric series. We obtain immdediately the theorem:
The series is absolutely
convergent, if from a certain term onwards there holds a relation
of the form
where c is a positive number independent of n and q is any fixed positive number less than 1.
This test is usually expressed in one of the following, weaker forms:
The series converges absolutely,
if there holds from a certain term onwards a relation of the form
where q is again a positive number less than 1 and independent of n, or, if from a certain term onward there holds a relation of the form
where q is again a positive number less than 1. In particular, the conditions of these tests are satisfied if there applies a relation of the form
These statements are easily established in the following manner:
Let the criterion IIa, the ratio
test, be satisfied from the
suffix n0 onward, i.e., when n > n0.
For the sake of brevity, let 
and find that
etc.; hence
which establishes our statement. For the Criterion IIb - the root test -we have at once |an| < qn, and our statement follows immediately.
Finally, in order to prove III, we consider an
arbitrary number q such that k < q
< 1. Then, from a certain n0 onwards,
i.e.,, when n > n0, it is certain
that 
or
as
the case may be, since from a certain term onwards the values of 
differ
from k by less than (g - k). The
statement is then established by reference to the results already
proved.
We stress the point that the four tests derived from the original criterion |an|<cqn are not equivalent to each other or to the original one, i.e., that they cannot be derived from one another in both directions. We shall soon see from examples that, if a series satisfies one of these conditions, it need not by any means satisfy all the other ones.
* More exactly: If IIIa is fulfilled, then IIa ie fulfilled: if IIIb, then IIb; if IIIa, then IIIb; if IIa, then IIb, and if any of the four is satisfied, then so is I. None of these statements can he reversed.
For the sake of completeness, it may be pointed out that a series certainly diverges if from a certain term onwards
for a properly chosen positive number c or if from a certain term onwards
or if
where k > 1. In fact, as we immediately recognize, in such a series the terms cannot tend to zero as n increases, whence the series must diverge. (Under these conditions, the series cannot even be conditionally convergent.)
Our tests furnish sufficient conditions for the absolute convergence of a series, i.e., if they are satisfied, we can conclude that the series converges absolutely. However, they are definitely not necessary conditions, i.e., absolutely convergent series can be found which do not satisfy these conditions.
For example, the knowledge that
does not entitle us to make any statement about the convergence of the series. Such a series may converge or diverge. For example, the series
for which 
is divergent, as we have seen
before. On the other hand, we shall soon see that the series 
which
satisfies the same relations, converges.
As an example of the application of our tests, we first consider the series
For this series,
It follows from the ratio test as well as from the root test even in the weaker form III that the series converges, if |q| < 1.
On the other hand, if we consider the series
we can no longer prove convergence by the ratio
test when ½ £ |q| < 1, because then |2q2n+1/q2n|
= 2|q| ³ 1. But the root test yields immediately 
and shows
that the series converges provided |q| < 1, which, of
course, we could have observed directly.
8.2.3 Comparison with an Integral:
We now proceed to a discussion of convergence which is independent of the preceding considerations. (cf. A7.1) We shall carry it out for the particularly simple and important case of the series
where the general term an
is 1/na, a > 0. In order to investigate the properties 
of this series, we consider the graph of the function y
= 1/xa and mark off on the x-axis the integral values
x = 1, x = 2, ···. We first construct the
rectangle of height 1/na over
the interval n-1 £ x £ n of the x-axis (n > 1)
and compare it with the area of the region bounded by the same
interval of the x -axis, the ordinates at the ends and
the curve y=1/xa (this
region is shown shaded in Fig. 2). Secondly, we construct the
rectangle of height 1/na lying
above the interval n£x£n+1 and similarly compare it with the area of
the region lying above the same interval and below the curve
(this region is cross-hatched in Fig. 2). In the first case, the
area under the curve is obviously larger than the area of the
rectangle; in the second case, it is less than the area of the
rectangle. In other words,
as we may also prove directly from the integral
itself (cf. 2.7.2). Writing down this
inequality for n = 2, n = 3, ··· , n
= m and summing, we obtain the estimate * for the m-th
partial sum 
* From this relation follows at
once that the sequence of numbers 
is bounded below.
Since we see from the inequality 
that the sequence is
monotonic decreasing, it must approach a limit
The number C with the value 0.577,2··· is called Euler's constant. In contrast to the other important special numbers of analysis, such as p and e, no other expression with a simple law of formation has been found for Euler's constant.
Now, as m increases, the integral 
tends to
a finite limit or increases without limit according to whether a > l or
a £ 1. Consequently,
the monotonic sequence of the numbers sm
is bounded or increases beyond all bounds according to whether a > or a £ 1, whence follows
the theorem:
The series
converges - and, of course, absolutely - if and only if, a > 1.
The divergence of the harmonic series, which we have proved previously in a different way, is an immediate consequence of this. In particular, we see that the series
converge.
The series 
, the convergence of
which we have just studied, frequently serve as comparison series
in investigations of convergence. For example, we see at once
that for a > 1 the series 
converges absolutely if the absolute
values |cn| of the coefficients remain less than a fixed bound
independent of n.
Find out whether the series 1. - 6. converge or do not converge:
Estimate the error after n terms of the series 7. - 10.:
11. Prove that 
converges.
12. Does 
converges.
13.* Prove that 
converges when a > 1 and
diverges when a £ 1.
14.* Prove that 
converges when a > 1 and
diverges when a £ 1.
15. Prove that if ui
³ 0 (i
= 1, 2, 3, ··· and 
and 
converge, then converges.
16. Show that if both 
S converge, then 
also
converges.
17. Prove that
18.* Prove that, if n is an arbitrary integer greater than 1,
where ann is defined as follows:
8.3 Sequences and Series of Functions
8.3.1 General Remarks: The terms of the infinite series hitherto considered have been constants, whence these series (when convergent) always represented definite numbers. But both in theory and in applications, the series of outstanding importance are those in which the terms are functions of a variable, so that the sum of the series is also a function of the variable, as in the case of Taylor series.
We shall therefore consider a series
in which the functions gn(x) are defined in an interval a £ x £ b. We will denote the n-th partial sum
by fn(x).
Then the sum f(x) of our series, where it
exists, is simply the limit 
We may therefore regard the sum of an infinite series of functions as the limit of a sequence of functions f1(x), f2(x), ··· , fn(x), ···. Conversely, for any such sequence of functions f1(x), f2(x), ···, we can form an equivalent series by setting g1(x) = f1(x) ···and gn(x) = fn(x) - fn-1(x) for n > 1. Hence, when it is convenient, we can pass from the consideration of series to that of sequences and conversely.
8.3.2 Limiting Processes with
Functions and Curves: We shall now
state exactly what we mean by saying that a function f(x)
is the limit of a sequence f1(x), f2(x),
··· , fn(x), ··· in
an interval a £ x £ b. The definition is: The sequence f1(x),
f2(x), ··· converges in that
interval to the limit function f(x), if at
each point x of the interval the values fn(x)
converge in the usual sense to the value f(x).
. In this case, we shall write 
According to Cauchy's test , we can express
the convergence of the sequence without necessarily knowing or
stating the limit function f(x). In fact, our
sequence of functions will converge to a limit function if, and
only if, at each point x in our interval and for every
positive number e the quantity |fn(x)
- fm(x)| is less than e, provided
that the numbers n and m are chosen large
enough, i.e., larger than a certain number N = N(e). This
number N(e) usually depends on e and x and increases beyond all bounds as e tends to
zero.
We have frequently encountered cases of limits of sequences of functions. We mention only the definition of the power xa for irrational values of a by the equation
where r1, r2, ···, rn, ··· is a sequence of rational numbers tending to a or the equation
where the functions fn(x) on the right hand side are polynomials of degree n.
The graphical representation of functions by
means of curves suggests that we can also speak of limits of sequences
of curves, saying, for
example, that the graphs of the above limit functions xa
and ex are to be regarded as the limit
curves of the graphs of the
functions 
respectively. However, there is a fine distinction
between the passages to the limit with functions and with curves.
Until the middle of the Nineteenth Century, this distinction was
not sufficiently recognized and only by having a clear idea of it
can we avoid apparent paradoxes. We shall illustrate this point
by an example.
Consider the
functions
in the interval 0 £ x £ 1. All these functions are continuous and the limit function
exists. However, this limit function is not continuous. On the contrary, since for all values of n the value of the function fn( 1) = 1, the limit
while, on the other hand, for 0 £ x <
I, the limit 
as we have seen in 1.5.6 . The function f(x)
is therefore a discontinuous function which at x=1 has
the value 1 and for all other values of x in the
interval has the value 0.
This discontinuity becomes clear, if we consider the graphs Cn of the functions y = fn(x). These (cf. Fig. 16 in 1.5.6) are continuous curves, all of which pass through the origin and the point x = 1, y = 1 and which draw in closer and closer to the x-axis as n increases. The curves possess a limit curve C which is not at all discontinuous, but consists (Fig. 3 above) of the portion of the x-axis between x=0 and x=1 and the portion of the line x=1 between y=0 and y=1. The curves therefore converge to a continuous limit curve with a vertical portion, while the functions converge to a discontinuous limit function. We thus recognize that this discontinuity of the limit function expresses itself by the occurrence in the limit curve of a portion perpendicular to the x-axis. Such a portion must involve a discontinuity in the limit function and, in fact, such a portion is always present when the limit function is discontinuous. This limit curve is not the graph of the limit function nor can any curve with a vertical portion be the graph of any single-valued function y = f(x); in fact, corresponding to the value of x at which the vertical portion occurs, the curve gives an infinite number of values of y and the function only one. Hence the limit of the graphs of the functions fn(x) is not the same as the graph of the limit f(x) of these functions.
Naturally, corresponding statements also apply to infinite series.