Sequences and Series

Sequences and limits
Note: 2.5 lectures
Analysis is essentially about taking limits. The most basic type of a limit is a limit of a sequence
of real numbers. We have already seen sequences used informally. Let us give the formal definition.
Definition 2.1.1. A sequence is a function x : N!R. Instead of x(n) we will usually denote the
nth element in the sequence by xn. We will use the notation fxng or more precisely
fxng¥
n=1
to denote a sequence.
A sequence fxng is bounded if there exists an B 2 R such that
jxnj  B
for all n 2 N. In other words, if the set fxn j n 2 Ng is bounded.
For example, f1=ng¥
n=1 stands for the sequence 1; 1=2; 1=3; 1=4; 1=5; : : :. We will usually just write
f1=ng. When we need to give a concrete sequence we will often give each term as a formula in
terms of n. The sequence f1=ng is a bounded sequence (B = 1 will suffice). On the other hand the
sequence fng stands for 1;2;3;4; : : :, and this sequence is not bounded (why?).
While the notation for a sequence is generally similar to that of a set, the notions are distinct.
For example, the sequence f(􀀀1)ng is the sequence 􀀀1;1;􀀀1;1;􀀀1;1; : : :. Whereas the set of
values, the range of the sequence, is just the set f􀀀1;1g. You could write this set as f(􀀀1)n j n 2 Ng.
When ambiguity could arise, we use the words sequence or set to distinguish the two concepts.
Another example of a sequence is the constant sequence. That is a sequence consisting of a
single constant c;c;c;c; : : : for some c 2 R.
[BS] use the notation (xn) to denote a sequence instead of fxng, which is what [R2] uses. Both are common.
39
40 CHAPTER 2. SEQUENCES AND SERIES
We now get to the idea of a limit of a sequence. We will see in Proposition 2.1.6 that the notation
below is well defined. That is, if a limit exists, then it is unique. So it makes sense to talk about the
limit of a sequence.
Definition 2.1.2. A sequence fxng is said to converge to a number x 2 R, if for every e > 0, there
exists an M 2 N such that jxn􀀀xj < e for all n  M. The number x is said to be the limit of fxng.
We will write
lim
n!¥
xn := x:
A sequence that converges is said to be convergent. Otherwise, the sequence is said to be
divergent.
It is good to know intuitively what a limit means. It means that eventually every number in the
sequence is close to the number x. More precisely, you can be arbitrarily close to the limit, provided
you go far enough in the sequence. It does not mean you will ever reach the limit. It is possible, and
quite common, that there is no xn in the sequence that equals the limit x.
When we write lim xn = x for some real number x, we are saying two things. First, that xn is
convergent, and second that the limit is x.
The above definition is one of the most important definitions in analysis, and it is necessary to
understand it perfectly. The key point in the definition is that given any e > 0, you can find an M.
The M can depend on e, so you only pick an M once you know e. Let us illustrate this concept on a
few examples.
Example 2.1.3: The constant sequence 1;1;1;1; : : : is convergent and the limit is 1. For every
e > 0, you can pick M = 1.
Example 2.1.4: The sequence f1=ng is convergent and
lim
n!¥
1
n
= 0:
Let us verify this claim. Given an e > 0, we can find an M such that 0 < 1=M < e (Archimedean
property at work). Then for all n  M we have that
jxn􀀀0j =

1
n

=
1
n

1
M
< e:
Example 2.1.5: The sequence f(􀀀1)ng is divergent. It is not hard to see. If there were a limit x,
then for e = 1
2 we expect an M that satisfies the definition. Suppose such an M exists, then for an
even n  M we compute
1=2 > jxn􀀀xj = j1􀀀xj and 1=2 > jxn+1􀀀xj = j􀀀1􀀀xj :
But
2 = j1􀀀x􀀀(􀀀1􀀀x)j  j1􀀀xj+j􀀀1􀀀xj < 1=2+1=2 = 1;
and that is a contradiction.
2.1. SEQUENCES AND LIMITS 41
Proposition 2.1.6. A convergent sequence has a unique limit.
The proof of this proposition exhibits a useful technique in analysis. Many proofs follow the
same general scheme. We want to show a certain quantity is zero. We write the quantity using the
triangle inequality as two quantities, and we estimate each one by arbitrarily small numbers.
Proof. Suppose that the sequence fxng has the limit x and the limit y. Take an arbitrary e > 0. From
the definition we find an M1 such that for all n  M1, jxn􀀀xj < e=2. Similarly we find an M2 such
that for all n  M2 we have jxn􀀀yj < e=2. Now take M := maxfM1;M2g. For n  M (so that both
n  M1 and n  M2) we have
jy􀀀xj = jxn􀀀x􀀀(xn􀀀y)j
 jxn􀀀xj+jxn􀀀yj
<
e
2
+
e
2
= e:
As jy􀀀xj < e for all e > 0, then jy􀀀xj = 0 and y = x. Hence the limit (if it exists) is unique.
Proposition 2.1.7. A convergent sequence fxng is bounded.
Proof. Suppose that fxng converges to x. Thus there exists a M 2 N such that for all n  M we
have jxn􀀀xj < 1. Let B1 := jxj+1 and note that for n  M we have
jxnj = jxn􀀀x+xj
 jxn􀀀xj+jxj
< 1+jxj = B1:
The set fjx1j ; jx2j ; : : : ; jxM􀀀1jg is a finite set and hence let
B2 := maxfjx1j ; jx2j ; : : : ; jxM􀀀1jg:
Let B := maxfB1;B2g. We then have that for all n 2 N
jxnj  B:
Example 2.1.8: The sequence
n
n2+1
n2+n
o
converges and
lim
n!¥
n2+1
n2+n
= 1:
42 CHAPTER 2. SEQUENCES AND SERIES
Given any e > 0, find M 2 N such that 1
M+1 < e. Then for any n  M we have

n2+1
n2+n
􀀀1

=

n2+1􀀀(n2+n)
n2+n

=

1􀀀n
n2+n

=
n􀀀1
n2+n

n
n2+n
=
1
n+1

1
M+1
< e:
Therefore, lim n2+1
n2+n = 1.