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By Cauchy's inequality, we have
\tam\At/Xx* Hence, using capital letters to denote absolute values,
T,no + 7mX + T,mX3 + ... + TmnXn + ... S e I X*[Rf.
But when X < /?, the series 2J"/^," is convergent and has the
sum Ri/(Rx—X); therefore 2 /m„jrB converges absolutely for
all values X< J?,. This means that it converges absolutely in the open region (R); for, if x be any selected point of this region, Rx can be chosen between X and R and the absolute convergence at x follows at once.
It results from the addition of (1) and (2) that the series
2 anxn converges within the circle (R).
III. Finally we have to prove that
%uq = %anxn within the circle (R).
Select any value x within (R) and then take Ri intermediate between X and R. For this value of x and for the same value of m as before, we have to prove that
m — l an oo 00
2 uq + 2 «, = 2 smnxn + 2 tmnxn;
j=0 q = m "=0 n=0
or removing the first terms on the two sides of the equation, since these are known to be equal, we have to prove that
2 uq = 2 tmnxn (3).
This can be done very readily by the use of the inequalities that have been found above. For
and whenever it can be asserted of any fixed quantity that its absolute value is less than an arbitrarily small positive number, that quantity is necessarily zero; hence (3) is proved.
We proceed to some important applications of the theorem.
82. Remarks on Weierstrass's Theorem. This theorem of Weierstrass's on series of power series in x can be extended at once to series of power series in i/x. It must be remembered that now the domain of convergence of a power series is the part of the plane exterior to some circle (o, R). The theorem as extended runs :—If the terms of the series £«,, where
uq = aq(t + aq,/x+... + aqn/xn + ..., converge outside the circle (R') and the series itself converges uniformly on the perimeter of every circle (Rf) where R( > R', then for every point x exterior to t/ee circle (/?') the series 2«, is expressible as a single power series in i/x.
Combining this result with that given earlier we obtain at
once a sufficient condition for the composition of a series of
+00 series 2 anx* into a single series of that form. Instead of a
region interior to (R) or a region exterior to (R'), we have the annular region (R', R) where R' is supposed less than R.
A convenient modification of Weierstrass's theorem replaces
the uniform convergence of the series 2 uq over every circle
(R,), where ^1 < R, by the uniform convergence of the series for the closed regions (Ri). Evidently the former condition is contained in the latter.
We shall, in general, use Weierstrass's criterion for the cpnversion of the double series 2«g into the single series Px, in preference to that of Cauchy (§ 69, corollary). There is nothing in either method to indicate whether the region for which the equation is proved to hold good includes all points that satisfy the equation Xuq=Px; later on we shall see that Taylor's theorem furnishes an example where the information furnished by these criteria,—exact of course as far as it goes,—proves to be incomplete.
At first sight Cauchy's criterion seems both simpler and
more effective than that of Weierstrass; that this is not always
the case appears from the series
x-i (x-1\* tx- i\n , .
i + -—+ +...+ + (1),
The series (1) converges for all values of x such that
i.e. for all values of x for which the real parts are less than
2; and it converges uniformly for all values of x such that
S a, where o is a proper fraction. The component series
have 3 for a common radius of convergence; hence all the conditions of Weierstrass's theorem are satisfied by taking Ji to be 2, and we see without further discussion that (2) can be expressed as a power series whose radius of convergence is at least as great as 2.
Cauchy's method uses the new series
3 + 3^+38^+-~3 + 3^" replacing in (1) by - H v?; and requires for the convergence of this new series the inequality
that is, X<3/2. It has given us therefore less information as
to the domain of convergence of the final series and has required
the summation of a new series.
The two series
x x' x* .
+ 7—Zi+^—li + (3).
I — X2 I — X* l—X*
in which X < 1, can be represented as double series by expanding the individual terms as power series; furthermore they arise from one and the same array
x x* xs x* ...
x' x" X3 X12 ...
by adding by rows and columns respectively. The double series is absolutely convergent, for the sum by rows of the absolute values of the separate terms is the convergent series
X X8 X"
i-X' i-X3^ i-X*^""
Hence Cauchy's criterion applies and the sum by columns must be equal to the sum by rows, and therefore the series (3), (4) have the same sum.
To see that Weierstrass's criterion applies to this case, observe that for any assigned value of x such that X < 1, we can take R between X and 1 and get
I — A n=0
£ a series which is uniformly convergent within (R). It follows that the series (1) is uniformly convergent within (R) and therefore comes under the operation of Weierstrass's theorem.
Ex. Prove that
x 2x2 yfi x Xs x3
i^x+ i-x* + T^3 + '"-(i-xY+ (i-x3)3 + (i^x°?+'"
83. Applications of Weierstrass's Theorem. (i) The
theorem that tlie product of two power series.
is itself a power series affords a simple example of Weierstrass's theorem on double series.
Suppose that Px, Qx both converge in the open region (R),
and let us consider the series ^.fvX, where f^= b^Px.
This series %f„x converges uniformly in the closed region (R^) where o < ^ <R; for it is possible to find a f i such that at every point x of this region (R^ we have, for values n S /*,
\rn(x)\<eu ex\Px\<e, where rn(x) is the remainder of Qx after n terms. We may add therefore by columns and thus get
Px x Qx = 2 (ajbn + «,£„-, + «.A-3 + • • • + a«A) •*"
for all points of the open region (R).
In particular (Px)3 can be arranged as a power series in x which is convergent when Px itself is; and the same can be said of any positive integral power of Px.
Ex. Show that if all three series
converge, then the third is the product of the first two.
(ii) A sufficient condition that P (Px) shall be expressible as Px. Suppose that Px is a power series with a radius of conver
gence R, and that 2«njn is a power series with a radius of
convergence R2; then 2 an (Px)n is not to be treated as a power "=o
series in x without a preliminary examination of the circumstances of the case. It is desirable to have a sufficient condition that
the transformation y = Px shall convert the series 2 a„yn into Px.
First let us simplify matters by making the series in y a terminating power series. We have then a finite number of power series in x associated with a-j, a^y\ a3y3 etc. and the sum of a finite number of power series is itself a power series. Thus a polynomial in y will in all cases transform into a power series in x.
But when we pass to the general case of an infinite series in y, we have the sum of an infinite number of power series in x and such a sum is not necessarily expressible as a power series. This case we shall now consider.
It is evident that if 2 an(Px)n is to be represented as a power "=o