SEQUENCES
( Jana Alexanderčíková, Veronika Schwarczová )
Definition: A sequence is each function f with D(f)⊆N0.
Notation: For sequences the notation x1, x2, xn is used instead of the usual notation x(1), x(2),
x(n) for functions.
x = xn nT T - does not have to be the whole set N0
but it has to be a subset of N0
10
y = y n n0 means a sequence with members y0, y1, …,y10
BASIC NOTIONS.
1. member ( term ) of the sequence xn nT
for example [4,3]  x means: x4=3 ( x4=3 – member, 4 – argument, 3 – value )
Remark: The number of values does not correspond with the number of members.
2. ways of determination of a sequence → according to functions ( sets of ordered pairs )
a) listing of all elements → ordered pairs, members
– can be used only for finite sequences
b) characteristic property of all elements ( members ) – often a formula
1) explicit – for the nth member of the sequence it is not necessary to know
the preceding members
for example yn = 2n+1
2) recursive – for the nth member of the sequence it is necessary to know
preceding members
for example Fn+1 = Fn+Fn-1, F1 = F2 = 1
FINITE SEQUENCES. INFINITE SEQUENCES.
Definition: A sequence is finite if its domain is a finite set ( if and only if a sequence has
finitely many members ). A sequence is infinite if it is not finite.
GRAPH OF THE SEQUENCE.
- graph of the function: a set of isolated points
-1-
PROPERTIES OF SEQUENCES.

Monotone sequences ( a special case of monotone functions )
→ increasing function f: (x1, x2)( x1<x2  f(x1)<f(x2) ), therefore
increasing sequence a: (n1, n2)( n1<n2  a n1 < a n2 )


→ consequence: for each sequence a n nT the following holds:
an nT is increasing (decreasing) if and only if (n)(an<(>)an+1)
Bounded sequence, maximum and minimum of a sequence
( again according to functions )
→ KR is an upper ( lower ) bound of the sequence a: (n)(an()K)
→ ak = max a n nT : (n)(anak), ak = min a n nT : (n)(anak)
→ observation: if the sequence a n nT is increasing ( decreasing ) then
min (max) an = a1 ( clearly the converse implications are not true in general )
→ observation: if the sequence a n nT is finite then it has its maximum and
minimum
Periodic sequence

→ definition: a sequence a n n  0 (1) is periodic if there exist n0N0 and kN such
that for each n>n0 the following holds: an+k=an.

The least k with this property is the period of a n n  0 (1) .

→ remark: the period of a constant sequence is 1.
Sequences cannot be neither even nor odd.
LIMIT OF A SEQUENCE.
( This notion refers only to infinite sequences, therefore the domain of indeces T is assumed
to be infinite.)
Definition: Let a n nT be a sequence of real numbers, let aR be a given number.
a = lim a n if (>0) (n0()) (n>n0) ( an( a-, a+ ) ).
n 
a according to this definition is a proper limit of a sequence.
-2-
Definition ( improper limit ): Let a n nT be a sequence of real numbers.
lim a n =  if (KR) (n0(K)) (n>n0) ( an( K,) ).
n 
lim a n = - if (KR) (n0(K)) (n>n0) ( an( -,K) ).
n 
Definition: A sequence is convergent if there exists aR which is the limit of this sequence.
A sequence is divergent if it is not convergent.
Consequence: A divergent sequence either has an improper limit or has no limit.
PROPERTIES OF LIMITS OF SEQUENCES.
Theorem: Each sequence has at most one limit.
Proof: by contradiction:
If there exists a sequence xn nT with at least two proper limits a<b:
lim x n = a  lim x n = b, therefore
n 
n 
(>0)(n0())(n>n0)( xn( a-; a+ ) )  (>0)(n0’())(n>n0’)( xn( b-; b+ ) )
b-a
If 0<<
then all members of the sequence xn nT for n>n0() are in the interval
2
( a-, a+ ) and all members for n>n0’() of the same sequence are in the interval
( b-, b+ )  a contradiction, since ( a-, a+ )  ( b-, b+ ) = .
The case of improper limit(-s) is analogical.
-3-
an
b
k
, an n , an , k an
bn
( kR … given constant ) under the conditions that all required limits exist and
all the terms are defined.
Theorem: For arbitrary sequences a n nT , bn nT the following holds:
if (n0)(n>n0)(anbn) and if a = lim a n , b = lim bn then ab.
Theorem: Limits preserve the operations a n , kan, an+bn, an-bn, anbn,
n 
n 
Warning: the implication (n)(an<bn)  a<b is not true in general!
∀n  n 0 : a n ≤bn ≤c n 


∧



Consequence: ∃lim a n  a,∃lim c n  c   (  b = lim bn  abc )
n 
n 
n 


∧

a  c



Observation: If lim a n = a and if (n0)(n>n0)(bn=an) then  lim bn = b and b=a.
n 
n 
If lim a n = a and if kZ, n0N such that (n>n0)(bn=an+k)
n 
then  lim bn = b and b=a.
n 
Observation: If lim a n = a and if bn nT ' is any infinite subsequence of a n nT
n 
then  lim bn = b and b=a.
n 
-4-
Theorem: 1. If n: an=c then lim a n = c
n 
2. lim
n 
1
n
=0
3. lim n = 
n 
1
an
4. lim (1  ) = e, if lim a n = 0 then lim (1  a n ) = e ( for an0 )
n 
1 n
n
n 
n 
5. lim n n = 1
n 
“Calculus with ” ( defined and undefined terms, for a,b,cR, 0<a, -1<c<1<b ):
Defined:
Undefined:
+a = , –a = , + = 
–
a = , (-a) = -,  = 
0
∞
∞
∞
= ,
= -
a
∞
(a)
a
a
a a 0
= ,
= -
,
,


0 0 0
0
0
∞
∞
∞
= ,
= -

0
0
0
a
a
0
= 0,
= 0,
=0


∞
a = , (-a) = 0,  = 
0
b = ,
c = 0, 0 = 0
1, 00
etc.
etc.
Remark: All these properties are the abbreviations for strictly finitary formulas
( formulas using no “infinity” ), namely formulas expressing particular properties
of sequences.
Observation: If a n nT is bounded and if lim bn = 0 then lim (a n  bn ) = 0.
n 
n 
an
= 0.
n  b
n
If a n nT is bounded and if lim bn =  then lim
n 
APPENDIX: CLUSTER POINT.
Definition: Let a n nT be a sequence of real numbers, let aR be a given number.
a is a cluster point of a n nT if ( infinitely many n)( an( a-, a+ ) ).
( improper cluster point , -…analogically )
-5-
Theorem: a is a cluster point of a n nT   a subsequence of a n nT with the limit a.
A sequence has at least two distinct cluster points if and only if it has no limit.
If a sequence has a limit then the limit is the unique cluster point of this sequence.
Each sequence has at least one cluster point ( including improper cluster points ).
ARITHMETIC SEQUENCE.
( k )
( k )
Definition: Let an n1 be a sequence of real numbers. an n1 is arithmetic if dR such
that (n)( an+1 = an+d ). This dR is the difference of an n1 .
( k )
Observation: an n1 is an arithmetic sequence  (dR)(n)( d = an+1 – an ).
Summary: recursive formula
an+1 = an+d
explicit formula
an = a1+(n-1)d
formula for 2 members
an – am = (n-m)d
a  an1 ank  an k
arithmetic average
an = n1

2
2
a  an
2a  (n  1)  d
sum of first n members
sn = 1
n  1
n
2
2
( k )
GEOMETRIC SEQUENCE.
( k )
( k )
Definition: Let an n1 be a sequence of real numbers. an n1 is geometric if qR such
that (n)( an+1 = anq ). This qR is the quotient of an n1 .
( k )
Observation: an n1 with all an0 is a geometric sequence  (qR)(n)( q =
( k )
Summary: recursive formula
explicit formula
formula for 2 members
geometric average
a n 1
).
an
an+1 = anq
an = a1qn-1
an
= qn–m ( for all ak0 )
am
an =
a n1  an1  ank  an k
1 qn
for q1, sn = a1n for q=1
1 q
Observation: Each arithmetic sequence with the difference d is a subset of some linear
function with the slope d.
Each geometric sequence with the quotient q is a subset of some real multiple
of the exponential function qx.
sum of first n members
sn = a1 
-6-
INFINITE SERIES. GEOMETRIC SERIES.

Definition: Let an n1 be a sequence of ( real ) numbers.
sn = a1+a2+…+an is the nth partial initial sum.

a
The infinite series
n 1
has the sum s if s = lim s n .
n
n 

a
Definition: A series
n 1
is convergent if it has a finite sum sR.
n

a
A series
n 1
n
is divergent if it is not convergent.
Consequence: A divergent series has the sum , or -, or no sum.
Theorem ( necessary condition for convergence of an infinite series ):

For each series
a
n 1

n
a
: if
n 1
has a finite sum sR then lim a n = 0.
n
n 
Warning: The converse implication is not true in general ( the condition is not sufficient! )!

Definition: Let for all nN fn be functions.The convergence region C of the series
f
n 1

is the set of all x  D ( f n ) such that the ( numerical ) series
n 1

Definition: Let
a
n 1

be an infinite series.
n
a
n 1
n
( x)

f
n 1
n
( x) converges.
is a geometric series if an n1 is

n
a geometric sequence.
 a

Observation: a geometric series in general:
n 1

n 1
, sn = a1 
1 q
1 qn
for q1
1 q

 1 q 
a
a
  1  lim 1  q n  1  1  lim q n
s lim s n  lim  a1 
n
n
n
1  q  1  q n
1 q

... for -1<q<1
lim q n = 0

n


n
... for q=1 ( but in this case sn = a1n, s = a1 lim n )
=1
n 
=
= no limit
Therefore
 a

s=
n 1
1

... for q>1
... for q-1
a1
1 q
= a1
= no sum
=0
 q n 1 =
... for -1<q<1
... convergent
... for a10, q1
... for a10, q-1
... for a1=0
... divergent
... divergent
... convergent ( trivial )
Theorem ( necessary and sufficient condition for convergence of a geometric series ):
 a

A geometric series
n 1
Then s =
1

 q n 1 with a10 is convergent  -1<q<1.
a1
.
1- q
-7-