A8.1 Multiplication and Division of Series
A8.1.1 Multiplication of Absolutely Convergent Series: Let
be two absolutely convergent series. Together with these series, consider the corresponding series of absolute values
Moreover, let
We now assert that the series 
is
absolutely convergent and that its sum is equal to A·B.
In order to prove this, we write down the series
the n²-th partial sum of which is AnBn,
and we assert that it converges absolutely. In fact, the partial
sums of the corresponding series with absolute values increase
monotonically; the n²-th partial sum is equal to 
which is
less than 
(and which tends to ). The series with
absolute values therefore converges and the series written down
above converges absolutely. The sum of the series is obviously AB,
since its n²-th partial sum is AnBn,
which tends to AB as n®¥. We now interchange the
order of the terms, which is permissible for absolutely
convergent series, and bracket successive terms together. In a
convergent series, we may bracket successive terms in as many
places as we desire without disturbing the convergence or
altering the sum of the series, because, if we bracket, say, the
terms
then, if we form the partial sum, we shall omit
those partial sums which originally fell between sn
and sm, which does not affect the
convergence or change the value of the limit. Also, if the series
was absolutely convergent before the brackets were inserted, it
remains absolutely convergent. Since the series
is formed in this way from the series written down above, the required proof is complete.
A8.1.2 Multiplication and Division of Power Series: The principal use of our theorem is found in the theory of power series. The following assertion is an immediate consequence of it:
The product of the two power series
is represented in the interval
of convergence common to the two power series by a third power
series 
the coefficients of which are given by
As regards the division of power series, we can
likewise represent the quotient of the two power series above by
a power series 
provided b0, the constant term
in the denominator, does not vanish. (In the latter case, such a
representation is, in general, not possible; indeed, it could not
converge at x=0 on account of the vanishing of the
denominator, while, on the other hand, every power series must
converge at x = 0.) The coefficients of the power series
can be calculated by recalling that
so that the following equations must be true:
We find readily from the first of these
equations q0, from the second one q1,
from the third one (by using the values q0
and q1) the value of q2,
etc. In order to strictly justify the expression of the quotient
of two power series by the third power series, we must still
investigate the convergence of the formally calculated power
series the general investigation of which we will omit. We
shall be content with the statement that the series for the quotient does
actually converge, provided x remains within a
sufficiently small interval, in which the denominator does not
vanish and both the numerator and the denominator are convergent
series.
A8.2 Infinite Series and Improper integrals
The infinite series and the concepts developed
in connection with them have simple applications and analogies in
the theory of improper integrals.
We shall confine ourselves here to the case of a convergent
integral with an infinite interval of integration, say an
integral of the form 
If we subdivide the interval of
integration by a sequence of numbers x0 = 0, x1,
··· , which tends monotonically to ¥, we can write the improper
integral in the form
where each term of our infinite series is an integral:
etc. This is true no matter how we choose the points xn , whence we can reduce the idea of a convergent improper integral to that of an infinite series in many ways.
It is especially convenient to choose the
points xn so that the integrand does not change sign within any
individual sub-interval. The series 
will then correspond to
the integral of the absolute value of our function
We are thus led naturally to the concept: An improper integral 
is said to be
absolutely convergent, if there exists the integral Otherwise, if our integral exists at all, we shall say
that it is conditionally convergent.
Some of the integrals considered earlier (4.8.4) such as
are absolutely convergent. On the other hand, the integral
studied in 4.8.5, is a simple example of a conditionally convergent integral. In order to give a proof of the convergence of this integral, which is independent of the former proof, we subdivide the interval from 0 to A at the points xn=np (n = 0, 1, 2, ··· , mA), where mA is the largest possible integer for which mAp£A. Hence we divide the integral into terms of the form
and a remainder RA of the form
It is clear that the quantities an have alternating signs, since sin x is alternately positive and negative in consecutive intervals. Moreover, |an+1| < |an|; indeed, on applying the transformation x = x - p, we have
Hence we see, by Leibnitz's test , that San converges. Moreover, the remainder RA has the absolute value
and this tends to 0 as A increases. Thus, if we let A tend to ¥ in the equation
the right hand side tends to San as a limit and our integral is convergent. However, the convergence is not absolute, because
In the introduction to this chapter (8.1), we drew attention to the fact that infinite series are only one way, although a particularly important way of representing numbers or functions by infinite processes. As an example of another such process, we shall introduce statements on infinite products without proofs.
We have dealt with Wallis' product
in which the number p/2 is expressed as an infinite product. We mean by the value of the infinite product
the limit of the sequences of partial products
provided it exists.
Naturally, the factors a1, a2. a3, ··· may also be functions of a variable x. An especially interesting example is the infinite product for the function sin x
which we shall obtain in 9.4.8.
The infinite product for the zeta function has a very important role in the theory of numbers. In order to retain the notation, used in number theory, we denote here the independent variable by s and define the zeta function for s > 1 by
We know from 8.2.3 that the series on the right-hand side converges if s > 1. If p is any number larger than 1, we obtain the equation
by expanding the geometric series. If we imagine this series written down for all the prime numbers p1, p2, p3, ··· in increasing order of magnitude and all the equations thus formed multiplied together, we obtain on the left-hand side a product of the form
If, without stopping to justify the process in any way, we multiply together the series on the right-hand sides of our equations and, in addition, recall that by an elementary theorem each integer n > 1 can be expressed in one and only one way as a product of powers of different prime numbers, we find that the product on the right hand side is again the function z (s), whence we obtain the remarkable product form
This product form, the derivation of which we have only sketched here briefly, is actually an expression of the zeta function as an infinite product, since the number of prime numbers is infinite.
In the general theory of infinite products, we usually exclude the case where the product a1·a2·a3 ··· has the limit zero, whence it is especially important that none of the factors an should vanish. In order that the product may converge, the factors an must accordingly tend to 1 as n increases. Since we can, if necessary, omit a finite number of factors (this has no bearing on the question of convergence), we may assume that an > 0. This case is the topic of the theorem:
A necessary and sufficient
condition for the convergence of the product 
where an > 0 is that the series 
should
converge.
Indeed, it is clear that the partial sums 
of this
series will tend to a definite limit, if and only if the partial
products a1a2a3···an
possess a positive limit.
In studying convergence, we usually apply the following criterion (a sufficient condition), where we put an = 1 + an :
The product
converges, if the series
converges and no factor (1 + an) is zero.
In the proof, we may assume, if necessary after omission of a finite number of factors, that each |an| < 1/2. We have then 1 - |an| > 1/2. By the mean value theorem,
for 0 < q < 1, whence
and so the convergence of the series 
follows
from the convergence of .
It follows from our criterion that the infinite product, given above for sin p x, converges for all values of x except for x = 0, ± 1, ± 2 ···, where factors of the product are zero. Moreover, for p ³ 2 and s > 1, we readily find that
Now, if we let p assume all prime
values, the series 
must converge, since the terms form only
a part of the convergent series 
. The convergence of the
product 
for s > 1 has thus been proved.
A8.4 Series involving Bernoulli Numbers
So far, we have given no expansions in power series for certain elementary functions, e.g., tan x. The reason is that the numerical coefficients which occur are not of any very simple form. We can express these coefficients and those in the series for a number of other functions in terms of the solaced Bernoulli numbers. These numbers are certain rational numbers with a not very simple law of formation which occur in many parts of analysis. We arrive at them most simply by expanding the function
in a power series of the form
If we write this equation in the form
and substitute on the right hand side the power series for ex - 1, we obtain, as in A8.1.2, a recurrence relation which allows the computation of all the numbers Bn . These numbers are called Bernoulli numbers.
In some publications, a slightly different notation is employed with the basic formula
)
They are rational, since in their formation only rational operations are used; as we easily verify, they vanish for all odd indices other than n = 1. The first few Bernoulli numbers are
We must content ourselves with a brief hint as to how these numbers are involved in the power series under consideration. Firstly, the transformation
yields
If we replace x by 2x, we obtain
for |x| < p, after replacing x by - x,
Now, using the equation 
, we find the series
valid for |x| < p /2.
For further information, we refer the reader, for example, to the more detailed treatise of K. Knopp, Theory and Application of Infinite Series, p. 183, (Blackie & Son, Ltd., 1928.
1. Prove that the power series for 
still
converges when x = 1.
2. Prove that there is for every positive e a
polynomial in x which represents in the interval 0 £ x £ 1 with an error
less than e.
3. Prove that there is for every positive e a polynomial in t which represents |t| in the interval -1 £ t £ l with an error less than e.
4. Weierstrass' Approximation Theorem: Prove that, if f(x) is continuous in a £ x £ b, then there exists for every positive e a polynomial P(x) such that |f(x)-P(x)| < e for all values of x in the interval a £ x £ b.
5. Prove that the following infinite products converge:
6. Prove by the above methods that 
diverges.
7. Use the identity
(where pi is the i-th prime number) to prove that the number of primes is infinite.
8. Prove the identity
for |x| < 1.