Power series

4.4 OPERATIONS WITH SERIES
is always less than uN for all m and un → 0 as n → ∞, the alternating series
converges. It is clear that an analogous proof can be constructed in the case
where N is even.
Determine whether the following series converges:
∞
n=1
(−1)n+1
1
1 1
= 1 − + − ··· .
n
2 3
This alternating series clearly satisfies conditions (i) and (ii) above and hence converges.
However, as shown above by the method of grouping terms, the corresponding series with
all positive terms is divergent. 4.4 Operations with series
Simple operations with series are fairly intuitive, and we discuss them here only
for completeness. The following points apply to both finite and infinite series
unless otherwise stated.
(i) If
u = S then
ku = kS where k is any constant.
n
n
vn = T then (un + vn ) = S + T .
(ii) If
un = S and
(iii) If
un = S then a + un = a + S. A simple extension of this trivial result
shows that the removal or insertion of a finite number of terms anywhere
in a series does not affect its convergence.
vn are both absolutely convergent then
(iv) If the infinite series
un and
the series
wn , where
wn = u1 vn + u2 vn−1 + · · · + un v1 ,
is also absolutely convergent. The series
wn is called the Cauchy product
of the two original series. Furthermore, if
un converges to the sum S
wn converges to the sum ST .
and
vn converges to the sum T then
(v) It is not true in general that term-by-term differentiation or integration of
a series will result in a new series with the same convergence properties.
4.5 Power series
A power series has the form
P (x) = a0 + a1 x + a2 x2 + a3 x3 + · · · ,
where a0 , a1 , a2 , a3 etc. are constants. Such series regularly occur in physics and
engineering and are useful because, for |x| < 1, the later terms in the series may
become very small and be discarded. For example the series
P (x) = 1 + x + x2 + x3 + · · · ,
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SERIES AND LIMITS
although in principle infinitely long, in practice may be simplified if x happens to
have a value small compared with unity. To see this note that P (x) for x = 0.1
has the following values: 1, if just one term is taken into account; 1.1, for two
terms; 1.11, for three terms; 1.111, for four terms, etc. If the quantity that it
represents can only be measured with an accuracy of two decimal places, then all
but the first three terms may be ignored, i.e. when x = 0.1 or less
P (x) = 1 + x + x2 + O(x3 ) ≈ 1 + x + x2 .
This sort of approximation is often used to simplify equations into manageable
forms. It may seem imprecise at first but is perfectly acceptable insofar as it
matches the experimental accuracy that can be achieved.
The symbols O and ≈ used above need some further explanation. They are
used to compare the behaviour of two functions when a variable upon which
both functions depend tends to a particular limit, usually zero or infinity (and
obvious from the context). For two functions f(x) and g(x), with g positive, the
formal definitions of the above symbols are as follows:
(i) If there exists a constant k such that |f| ≤ kg as the limit is approached
then f = O(g).
(ii) If as the limit of x is approached f/g tends to a limit l, where l = 0, then
f ≈ lg. The statement f ≈ g means that the ratio of the two sides tends
to unity.
4.5.1 Convergence of power series
The convergence or otherwise of power series is a crucial consideration in practical
terms. For example, if we are to use a power series as an approximation, it is
clearly important that it tends to the precise answer as more and more terms of
the approximation are taken. Consider the general power series
P (x) = a0 + a1 x + a2 x2 + · · · .
Using d’Alembert’s ratio test (see subsection 4.3.2), we see that P (x) converges
absolutely if
an+1 an+1 < 1.
x = |x| lim ρ = lim n→∞
n→∞
an
an Thus the convergence of P (x) depends upon the value of x, i.e. there is, in general,
a range of values of x for which P (x) converges, an interval of convergence. Note
that at the limits of this range ρ = 1, and so the series may converge or diverge.
The convergence of the series at the end-points may be determined by substituting
these values of x into the power series P (x) and testing the resulting series using
any applicable method (discussed in section 4.3).
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4.5 POWER SERIES
Determine the range of values of x for which the following power series converges:
P (x) = 1 + 2x + 4x2 + 8x3 + · · · .
By using the interval-of-convergence method discussed above,
n+1 2
ρ = lim n x = |2x|,
n→∞
2
and hence the power series will converge for |x| < 1/2. Examining the end-points of the
interval separately, we find
P (1/2) = 1 + 1 + 1 + · · · ,
P (−1/2) = 1 − 1 + 1 − · · · .
Obviously P (1/2) diverges, while P (−1/2) oscillates. Therefore P (x) is not convergent at
either end-point of the region but is convergent for −1 < x < 1. The convergence of power series may be extended to the case where the
parameter z is complex. For the power series
P (z) = a0 + a1 z + a2 z 2 + · · · ,
we find that P (z) converges if
an+1 an+1 < 1.
ρ = lim z = |z| lim n→∞
n→∞
an
an We therefore have a range in |z| for which P (z) converges, i.e. P (z) converges
for values of z lying within a circle in the Argand diagram (in this case centred
on the origin of the Argand diagram). The radius of the circle is called the
radius of convergence: if z lies inside the circle, the series will converge whereas
if z lies outside the circle, the series will diverge; if, though, z lies on the circle
then the convergence must be tested using another method. Clearly the radius of
convergence R is given by 1/R = limn→∞ |an+1 /an |.
Determine the range of values of z for which the following complex power series converges:
P (z) = 1 −
z
z2
z3
+
−
+ ··· .
2
4
8
We find that ρ = |z/2|, which shows that P (z) converges for |z| < 2. Therefore the circle
of convergence in the Argand diagram is centred on the origin and has a radius R = 2.
On this circle we must test the convergence by substituting the value of z into P (z) and
considering the resulting series. On the circle of convergence we can write z = 2 exp iθ.
Substituting this into P (z), we obtain
2 exp iθ
4 exp 2iθ
+
− ···
2
4
2
= 1 − exp iθ + [exp iθ] − · · · ,
P (z) = 1 −
which is a complex infinite geometric series with first term a = 1 and common ratio
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SERIES AND LIMITS
r = − exp iθ. Therefore, on the the circle of convergence we have
P (z) =
1
.
1 + exp iθ
Unless θ = π this is a finite complex number, and so P (z) converges at all points on the
circle |z| = 2 except at θ = π (i.e. z = −2), where it diverges. Note that P (z) is just the
binomial expansion of (1 + z/2)−1 , for which it is obvious that z = −2 is a singular point.
In general, for power series expansions of complex functions about a given point in the
complex plane, the circle of convergence extends as far as the nearest singular point. This
is discussed further in chapter 24. Note that the centre of the circle of convergence does not necessarily lie at the
origin. For example, applying the ratio test to the complex power series
P (z) = 1 +
(z − 1)3
z − 1 (z − 1)2
+
+
+ ··· ,
2
4
8
we find that for it to converge we require |(z − 1)/2| < 1. Thus the series converges
for z lying within a circle of radius 2 centred on the point (1, 0) in the Argand
diagram.
4.5.2 Operations with power series
The following rules are useful when manipulating power series; they apply to
power series in a real or complex variable.
(i) If two power series P (x) and Q(x) have regions of convergence that overlap
to some extent then the series produced by taking the sum, the difference or the
product of P (x) and Q(x) converges in the common region.
(ii) If two power series P (x) and Q(x) converge for all values of x then one
series may be substituted into the other to give a third series, which also converges
for all values of x. For example, consider the power series expansions of sin x and
ex given below in subsection 4.6.3,
x5
x7
x3
+
−
+ ···
3!
5!
7!
x3
x4
x2
ex = 1 + x +
+
+
+ ··· ,
2!
3!
4!
both of which converge for all values of x. Substituting the series for sin x into
that for ex we obtain
sin x = x −
3x4
8x5
x2
−
−
+ ··· ,
2!
4!
5!
which also converges for all values of x.
If, however, either of the power series P (x) and Q(x) has only a limited region
of convergence, or if they both do so, then further care must be taken when
substituting one series into the other. For example, suppose Q(x) converges for
all x, but P (x) only converges for x within a finite range. We may substitute
esin x = 1 + x +
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4.5 POWER SERIES
Q(x) into P (x) to obtain P (Q(x)), but we must be careful since the value of Q(x)
may lie outside the region of convergence for P (x), with the consequence that the
resulting series P (Q(x)) does not converge.
(iii) If a power series P (x) converges for a particular range of x then the series
obtained by differentiating every term and the series obtained by integrating every
term also converge in this range.
This is easily seen for the power series
P (x) = a0 + a1 x + a2 x2 + · · · ,
which converges if |x| < limn→∞ |an /an+1 | ≡ k. The series obtained by differentiating P (x) with respect to x is given by
dP
= a1 + 2a2 x + 3a3 x2 + · · ·
dx
and converges if
nan
= k.
|x| < lim n→∞ (n + 1)an+1 Similarly the series obtained by integrating P (x) term by term,
a2 x3
a1 x2
+
+ ··· ,
P (x) dx = a0 x +
2
3
converges if
(n + 2)an = k.
|x| < lim n→∞ (n + 1)an+1 So, series resulting from differentiation or integration have the same interval of
convergence as the original series. However, even if the original series converges
at either end-point of the interval, it is not necessarily the case that the new series
will do so. The new series must be tested separately at the end-points in order
to determine whether it converges there. Note that although power series may be
integrated or differentiated without altering their interval of convergence, this is
not true for series in general.
It is also worth noting that differentiating or integrating a power series term
by term within its interval of convergence is equivalent to differentiating or
integrating the function it represents. For example, consider the power series
expansion of sin x,
x5
x7
x3
+
−
+ ··· ,
(4.14)
3!
5!
7!
which converges for all values of x. If we differentiate term by term, the series
becomes
x4
x6
x2
+
−
+ ··· ,
1−
2!
4!
6!
which is the series expansion of cos x, as we expect.
sin x = x −
135