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Sequences and Series
§ 1.1 Sequences
In the fifth century B.C. the Greek philosopher Zeno of Elea posed a problem, now known as
Zeno’s paradox, concerns a race between the Greek hero Achilles and a tortoise that has been
given a head start. Zeno argued that Achilles could never pass the tortoise. Let Achilles starts at
position 1 L and the tortoise starts at position S1. When Achilles reaches the point L1 =S2 , the
tortoise is farther ahead at position S2. This process continues indefinitely and so it appears that
the tortoise will always be ahead.
One way of explaining this paradox is with the concept of a sequence. The successive positions
of Achilles {L1 , L2 , L3 ,…} and the successive positions of the tortoise {S1 , S2 , S3 ,…}
constitute what is known as sequences, generally denoted by {an }∞n =1 where an is a set of numbers
written in a definite order. In general, if the numbers an can be made as close as to the number, L
or the terms an approach the number L by taking n sufficiently large, then we use the concept of
the limit of a sequence, denoted as lim an = L . In view of Zeno’s paradox, the successive
n →∞
positions {L1 , L2 , L3 ,…} of Achilles and the successive positions {S1 , S2 , S3 ,…} of tortoise
constitute two sequences, where Ln <Sn for all n=1,2,..... Indeed, it can be shown that both
sequences have the same limit, i.e., lim Ln = lim Sn = L . It is precisely at this point L that Achilles
n →∞
n →∞
overtakes the tortoise.
Sequence: A sequence is a set of elements. The elements of the set can be either numbers or
letters or a combination of both. The elements of the set all follow the same rule (logical
progression). The number of elements in the set can be either finite or infinite. A sequence is
usually represented by using brackets of the form { } and placing either the rule or a number of
the elements inside the brackets. Some simple examples of sequences are listed below.
The alphabet: {a, b, c, ..., z}
The set of natural numbers less than or equal to 50: {1, 2, 3, 4, ..., 50}
The set of all natural numbers:
{1, 2, 3, ..., n, ...}
The set {an} where an = an−1 + 2, a1 = 1
NOTE: As this last example suggests, the general element of the sequence is normally
represented by using a subscripted letter, with the range of the subscript being the natural
(counting) numbers unless otherwise noted.
The two items of greatest interest with sequences are
(1) The representation of the general term of the sequence if not given, and
(2) What happens to the value of the general term as the value of the subscript increases.
The determination of the general term, when not given explicitly, can frequently be quite
challenging, because it rests primarily on pattern recognition.
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As the value of the subscript increases (tends to infinity) the general term of the sequence may or
may not have a single value. If it does have a single finite value, then the sequence is said to
converge, and the sequence converges to that single finite value. If the limiting value is infinite
or if no limiting value exists then the sequence is said to diverge.
Symbolically:
Given {an}, if lim a n = L where L < ∞ , then the sequence converges.
n →∞
Otherwise the sequence diverges.
Algebra of sequences:
Given any two sequences {an} with limit value A, {bn} with limit value B, and any two scalars k,
p, the following are always true:
(a)
(b)
(c)
(d)
{k an + p bn } is a convergent sequence with limit value kA + pB
{ an bn } is a convergent sequence with limit value AB
{ an / bn } is a convergent sequence with limit value A/B provided that B ≠ 0
if f (x) is a continuous function with lim f ( x ) = L , and if an = f (n) for all values of n
(e)
then {an} converges and has the limit value L
if an ≤ cn ≤ bn , then {cn} converges with limit value C where A ≤ C ≤ B
x→∞
Item (d) above permits us to use methods from the theory of functions, for example L’Hôpital’s
rule, and in item (e) above if the limit values of the sequences {an} and {bn} are both the same,
then this is called the squeeze theorem.
If each element of a sequence {an} is no less than all of its predecessors (a1 ≤ a2 ≤ a3 ≤ a4 ≤ ...)
then the sequence is called an increasing sequence.
If each element of a sequence {an} is no greater than all of its predecessors (a1 ≥ a2 ≥ a3 ≥ a4 ≥
...) then the sequence is called a decreasing sequence.
A monotonic sequence is one in which the elements are either increasing or decreasing.
If there exists a number M such that ⏐ an ⏐ ≤ M for all values of n then the sequence is said to
be bounded.
Theorem:
Every bounded monotonic sequence is convergent.
Standard Sequences:
Some of the most important sequences are
{r n } = {r1 , r 2 , r 3 , " } . This sequence converges whenever −1 < r ≤ 1.
(1)
{n r } = {1r , 2 r , 3r , " } . This sequence converges whenever r ≤ 0.
(2)
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Examples: Only two examples are presented here. They are examples of items (d), and (e) above.
Sample Problem 1:
⎧ ⎛ 1 ⎞⎫
⎪⎪ sin⎜ n ⎟ ⎪⎪
Find the limit of ⎨ ⎝ ⎠ ⎬ .
1
⎪
⎪
⎪⎩ n ⎪⎭
sin( x )
. We know from L’Hôpital’s Rule that as x approaches zero, the
x
function approaches the limit value of one. Hence, by item (d) above the sequence converges
and has the limit value of one.
Consider f ( x ) =
⎧ sin(n ) ⎫
Find the limit of ⎨
⎬.
⎩ n ⎭
Here we wish to use item (e) above as the squeeze theorem. It is easy to show that for every
1
sin(n )
1
value of n , −
≤
≤
, and that both the first and third sequences converge and
n
n
n
⎧ sin(n ) ⎫
that they both have the limit value of zero. Hence, it follows that ⎨
⎬ converges and also
⎩ n ⎭
has the limit value of zero.
Sample Problem 2:
§ 1.2 Series
Series: A series is a sum of elements. The sum can be finite or it can be infinite. The elements
of the series can be either numbers or letters or a combination of both. A series can be
represented
(a) by listing a number of elements along with the appropriate sign (+ or −) between the elements
OR
(b) by using what is called sigma notation with only the general term and the range of summation
indicated.
Examples:
(1)
1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + 9 − 10
10
n +1
(2)
∑n=1 (− 1) n
Both of these examples represent the same series.
As with sequences the main areas of interest with series are:
(a) the determination of the general term of the series if the general term is not given, and
(b) finding out whether or not the sum of the given series exists.
Again as with sequences the determination of the general term of the series, if the general term is
not given, relies heavily on pattern recognition.
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For a series that contains only finitely many terms, the sum always exists provided that each of
the terms of the series is finite.
For a series that contains infinitely many terms we need to use the following theorem.
Theorem: A series converges iff the associated sequence of partial sums represented by {Sk}
converges. The element Sk in the sequence above is defined as the sum of the first “k” terms
of the series.
In the remaining sections of this document, a number of different kinds of series will be
considered. They, generally speaking, fall into one of the following categories:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Telescoping series
Geometric series
Hyperharmonic series (also known as p-series)
Alternating series
Power series
Binomial series
Taylor series
We will also consider a number of tests that make it unnecessary to use the theorem mentioned
above. The various tests that will be studied are:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
nth term test (also known as the divergence test)
Geometric series test
Comparison tests
Alternating series test
Ratio test
Root test
For a series with both positive and negative terms it is necessary to consider two different kinds
of convergence. These are conditional convergence, and absolute convergence.
If a series contains only positive terms, then conditional convergence is impossible, and we
usually refer simply to convergence in this case.
Properties of Series:
1. Adding or deleting a finite number of finite terms in a given series has no effect
on the convergence of the given series.
2. If the series Σ an converges and has sum A, and if the series Σ bn converges and
has sum B, and if p and q are any finite constants, then Σ (pan + q bn)
converges and has sum (pA + qB).
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§ 1.3
Series Tests
In this section the various tests mentioned in the previous section will be introduced, and a
number of examples will be considered in class to illustrate the various tests.
General (nth) Term Test (also known as the Divergence Test):
∞
If lim a n ≠ 0 , then the series ∑n =1 a n diverges.
n→∞
NOTE:
This test is a test for divergence only, and says nothing about convergence.
Geometric Series Test:
A geometric series has the form
∑
∞
n =0
a r n , where “a” is some fixed scalar (real number).
A series of this type will converge provided that ⏐r⏐< 1, and the sum is
a
.
1− r
A proof of this result follows.
Consider the kth partial sum, and “r” times the kth partial sum of the series
Sk =
a + ar 1 + ar 2 + ar 3 + " + ar k
r S k = ar 1 + ar 2 + ar 3 + " + ar k + ar k +1
The difference between rSk and Sk is (r − 1)S k = a (r k +1 − 1) .
a (r k +1 − 1)
.
(r − 1)
Since the only place that “k” appears on the right in this last equation is in the numerator, the
limit of the sequence of partial sums {Sk} will exist iff the limit as k → ∞ exists as a finite
number. This is possible iff ⏐r⏐< 1, and if this is true then the limit value of the sequence of
a
partial sums, and hence the sum of the series, is S =
.
1− r
Provided that r ≠ 1, we can divide by (r − 1), to obtain
Sk =
Telescoping Series:
Generally, a telescoping series is a series in which the general term is a ratio of polynomials in
powers of “n”.
The method of partial fractions (learned when studying techniques of
integration) is normally used to rewrite the general term, and then the sequence of partial sums is
studied. This sequence will, most of the time, simplify to just a few terms, and the limit can then
be determined. One example of a telescoping series will be presented here, and additional
examples in class.
Sample Problem 3:
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Evaluate
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∑
∞
n =1
1
.
n +n
2
1
1
1
=
−
.
n +n
n
n +1
We now consider the partial sums S1, S2, S3, ..., Sn, ... until a pattern emerges and then the limit
value S will be determined.
1
1
1−
1−
=
S1 =
2
2
1
⎛ 1⎞ ⎛1 1⎞
1−
=
S2 =
⎜1 − ⎟ + ⎜ − ⎟
3
⎝ 2 ⎠ ⎝ 2 3⎠
1
⎛ 1⎞ ⎛1 1⎞
1−
=
S3 =
⎜1 − ⎟ + ⎜ − ⎟
4
⎝ 3⎠ ⎝ 3 4 ⎠
1
1
1
1
⎞
⎛
⎞ ⎛
1−
=
S4 =
⎜1 − ⎟ + ⎜ − ⎟
5
⎝ 4⎠ ⎝ 4 5⎠
#
#
#
1 ⎞
1
⎛ 1⎞ ⎛1
S n = ⎜1 − ⎟ + ⎜ −
⎟ = 1−
n +1
⎝ n ⎠ ⎝ n n + 1⎠
Since we have now determined the general pattern, the limit value S of the sequence of partial
sums, and hence the sum of the series is seen to have a value of “1”.
The general term an can be rewritten as a n =
2
Comparison Tests:
There are four comparison tests that are used to test series. There are two convergence tests, and
two divergence tests. In order to use these tests it is necessary to know a number of convergent
∞
series and a number of divergent series. For the tests that follow we shall assume that ∑n =1 cn
is some known convergent series, that
∑
∞
n =1
d n is some known divergent series, and that
∑
∞
n =1
an
is the series to be tested. Also it is to be assumed that for n ∈ {1, 2, 3, ..., (k−1)} the values of
an are finite, and that each of the series contains only positive terms.
Standard Comparison Tests:
Convergence Test:
If
∑ c is a convergent series and an ≤ cn for all n ≥ k,
then ∑ a is a convergent series.
If ∑ d is a divergent series and an ≥ dn for all n ≥ k,
then ∑ a is a divergent series.
∞
n =1 n
∞
n =1
Divergence Test:
n
∞
n =1 n
∞
n =1
n
Limit Comparison Tests:
Convergence Test:
If
∑
∞
an
= L
n →∞ c
n
c is a convergent series and lim
n =1 n
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where 0 ≤ L < ∞, then
If
Divergence Test:
∑
∞
n =1
∑
1
n =1
p
∞
∞
(− 1)
n =1
np
1
∑n
n =1
∑
∞
n =1
a n is a divergent series.
d is often the geometric series
n =1 n
1
∑n
p
∑
∞
n =0
a rn
.
converges absolutely when p > 1 and diverges otherwise.
A special case is the harmonic series
[The alternating p-series
∑
an
= L
dn
∞
c or
n =1 n
n =1
∑n
a n is a convergent series.
∑
∞
∞
∞
n =1
n →∞
or the hyperharmonic series (or p-series)
Property: The p-series
∞
d n is a divergent series and lim
where 0 < L ≤ ∞, then
The choice for the reference series
∑
=
1 1 1 1
+ + + + " , which diverges (p = 1).
1 2 3 4
n
converges absolutely when p > 1 ,
converges conditionally when 0 < p ≤ 1 and diverges otherwise.]
Alternating Series Test:
Given a series
∑
∞
n =1
a n = a1 + a2 + a3 + ... + a(k−1) +
∑
∞
n =k
an where a1 , a2 , a3 , ... , a(k−1) can be
a n +1
< 0 for all n ≥ k ,
an
if lim an = 0 , then the series converges. If lim an ≠ 0 , then the series diverges.
any finite real numbers, and
n →∞
n →∞
Ratio Test:
Given a series
that lim
n →∞
∑
∞
n =1
a n with no restriction on the values of the an’s except that they are finite, and
a n +1
= L , the series converges absolutely whenever 0 ≤ L < 1, diverges whenever
an
1 < L ≤ ∞, and the test fails if L = 1.
Root Test:
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∑
∞
Given a series
that lim ( a n
n →∞
n =1
)
1
n
a n with no restriction on the values of the an’s except that they are finite, and
= L , the series converges absolutely whenever 0 ≤ L < 1, diverges whenever
1 < L ≤ ∞, and the test fails if L = 1.
Absolute and Conditional Convergence:
A convergent series that contains an infinite number of both negative and positive terms must be
tested for absolute convergence.
∞
∞
A series of the form ∑n =1 a n is absolutely convergent iff ∑n =1 a n the series of absolute values
is convergent.
∞
If ∑n =1 a n is convergent, but
∑
∞
n =1
∑
∞
n =1
a n the series of absolute values is divergent, then the series
a n is conditionally convergent.
A shortcut:
In some cases it is easier to show that
∑
∞
a n is convergent.
n =1
It then follows immediately that the original series
§ 1.4
∑
∞
n =1
a n is absolutely convergent.
Power Series
Power Series: Any series of the form
∑
∞
n =0
cn ( x − a ) kn +b where b ∈ ⎥ and k = any non-zero real
number is called a power series. (It is a series in powers of (x−a), where “a” is called the centre
of the series).
A power series can be tested to determine absolute convergence by means of the ratio test,
introduced in the previous section. If we compare the terms of this series with the general term
∞
∞
of the series ∑n =0 a n , then we may set an = cn(x−a)kn+b. The
series
∑n=0 an converges
absolutely, by the ratio test, whenever lim
n →∞
cn(x−a)kn+b, this means that
cn +1 ( x − a ) k (n +1)+b
lim
n →∞
cn ( x − a ) kn +b
∑
∞
n =0
a n +1
= L , where 0 ≤ L < 1. Since we may set an =
an
cn ( x − a ) kn +b will converge absolutely whenever
cn +1 ( x − a ) k
= lim
n →∞
cn
This is equivalent to requiring that 0 ≤
= L , where 0 ≤ L < 1.
x−a
k
<
1
c
lim n +1
n →∞ c
n
(in the case k > 0).
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The series may converge absolutely, converge conditionally or diverge when
1
−1
or when ( x − a ) =
.
( x − a) =
1
1
k
k
c
c
lim n +1
lim n +1
n →∞ c
n →∞ c
n
n
These two points must be considered separately.
For all other values of | x−a| the series diverges.
The entire set of values of (x−a) for which the series converges (absolutely or conditionally) is
called the interval of convergence I. The radius of convergence R for the power series is
x−a
1
=
lim
n→∞
c( n +1)
1
k
= lim
n →∞
cn
c( n +1)
1
k
= R.
cn
One example, only, will be considered here, and a number of additional examples will be
considered in class.
Sample Problem 4:
Find the radius of convergence and the interval of convergence for the series
n
∞ ( x + 3)
∑n=0 n + 4 .
This series converges absolutely when
1
0 ≤ x+3 <
n+5
lim
n →∞ n + 4
i.e. 0 ≤ x + 3 <
1
The radius of convergence is R = 1.
1
which is a divergent series.
n =0
n+4
n
∞ (− 1)
which is a [conditionally] convergent
When (x + 3) = −1, the given series becomes ∑n =0
n+4
alternating series.
When (x + 3) = 1, the given series becomes
∑
∞
Hence, the series will converge whenever −1 ≤ x+3 < 1.
This can also be expressed by saying that the interval of convergence I for this series is
I = {x | −4 ≤ x < −2 }, or I = [−4, −2) .