MAT1032 Calculus II
Sum, Product and Quotient of Sequences, and
Further Important Theorems for Sequences
7th March, 2013
Engin MERMUT and Celal Cem SARIOĞLU
2013.03.07
Dokuz Eylül University Faculty of Science
Department of Mathematics İzmir/TURKEY
Contents
1 Sum, Product and Quotient of Convergent Sequences
1
2 Subsequences
3
3 Further Important Theorems for Sequences
3
1
Sum, Product and Quotient of Convergent Sequences
PROVE THE FOLLOWING THEOREMS FOR THE SUM, PRODUCT AND QUOTIENT
OF CONVERGENT SEQUENCES TILL NEXT WEEK.
∞
Proposition 1. Let (an )∞
n=1 and (bn )n=1 be sequences of real numbers and let L ∈ R. Then:
1. lim an = 0
⇐⇒
2. lim an = L
⇐⇒
3. lim an = L
=⇒
n→∞
n→∞
n→∞
lim |an | = 0.
n→∞
lim |an − L| = 0.
n→∞
lim |an | = |L|.
n→∞
4. If |an − L| ≤ bn for all n ∈ Z+ beyond some index N0 and lim bn = 0, then lim an = L.
n→∞
n→∞
∞
Question: Find a sequence (an )∞
n=1 that is divergent but (|an |)n=1 is convergent.
Proposition 2. Let (an )∞
n=1 be a sequence of real numbers, and let M ∈ R.
∞
If (an )n=1 converges to a real number L, then the sequence (M an )∞
n=1 converges to M L, that
is,
lim an = L ∈ R and M ∈ R
=⇒
lim M an = M L.
n→∞
n→∞
1
Product of a bounded sequence and a sequence that converges to 0
∞
Theorem 3. Let (an )∞
n=1 and (bn )n=1 be sequences of real numbers.
∞
∞
If (bn )n=1 is a bounded sequence and (an )∞
n=1 converges to 0, then the sequence (an bn )n=1
converges to 0, that is,
lim an = 0 and (bn )∞
n=1 is bounded
=⇒
n→∞
lim an bn = 0.
n→∞
Limit of Sum and Product of Convergent Sequences
∞
Let (an )∞
n=1 and (bn )n=1 be convergent sequences of real numbers. Say for real numbers L
∞
and M , (an )∞
n=1 converges to L and (bn )n=1 converges to M :
lim an = L
and
n→∞
lim bn = M.
n→∞
Let c ∈ R.
Then the sequences
(can )∞
n=1 ,
(an + bn )∞
n=1
(an bn )∞
n=1
and
are also convergent, and
1. lim can = cL,
n→∞
2. lim (an + bn ) = L + M , and
n→∞
3. lim (an bn ) = LM .
n→∞
Sequences with a nonzero limit
Theorem 4. If (an )∞
n=1 is a convergent sequence of real numbers with a nonzero limit, that is,
lim an = L ∈ R
L 6= 0,
and
n→∞
then there exists N0 ∈ Z+ such that for all n ∈ Z+ ,
n > N0
=⇒
an 6= 0
1
2
<
.
|an |
|L|
and
Corollary 5. If (an )∞
n=1 is a convergent sequence of real numbers with a nonzero limit, that is,
lim an = L ∈ R
L 6= 0,
and
n→∞
and if an 6= 0 for all n ∈ Z+ , then the sequence
1
an
∞
is a bounded sequence.
n=1
Quotient of a convergent sequence with a nonzero limit
Theorem 6. If (an )∞
n=1 is a convergent sequence of real numbers with a nonzero limit, that is,
lim an = L ∈ R
and if an 6= 0 for all n ∈
1
, that is,
L
Z+ ,
L 6= 0,
and
n→∞
then the sequence
1
an
∞
1
1
= .
n→∞ an
L
lim
2
is a convergent sequence with limit
n=1
Quotient of convergent sequences
∞
Let (an )∞
n=1 and (bn )n=1 be convergent sequences of real numbers. Say for real numbers L
∞
and M , (an )n=1 converges to L and (bn )∞
n=1 converges to M :
lim an = L
and
n→∞
If L 6= 0 and if an 6= 0 for all n ∈
with limit
Z+ ,
lim bn = M.
n→∞
then the sequence
M
, that is,
L
lim
n→∞
bn
an
∞
is a convergent sequence
n=1
M
bn
=
.
an
L
Comparison of Limits
∞
Let (an )∞
n=1 and (bn )n=1 be convergent sequences of real numbers. Say for real numbers L
∞
and M , (an )∞
n=1 converges to L and (bn )n=1 converges to M :
lim an = L
and
n→∞
lim bn = M.
n→∞
If an ≤ bn for all n ∈ Z+ beyond some index N0 , that is, if there exists N0 ∈ Z+ such that
an ≤ bn
for all integers n > N0 ,
then L ≤ M , that is,
lim an ≤ lim bn .
n→∞
2
n→∞
Subsequences
SEE THE LECTURE NOTES!
3
Further Important Theorems for Sequences
Further important theorems and concepts
Next year in your Analysis course, you shall see the following very important theorems and
concepts for sequences:
• Nested Interval Theorem
• Bolzano-Weierstrass Theorem
• Cauchy sequences
• A sequence of real numbers is a Cauchy sequence if and only if it is convergent. This is
indeed the Completeness property of R.
3
Nested Interval Theorem
Theorem 7. For every n ∈ Z+ , let In = [an , bn ] be a bounded closed interval, that is, an and bn
are real numbers such that an < bn for every n ∈ Z+ . Assume that (In )∞
n=1 is a nested sequence
+
of intervals, that is, In ⊇ In+1 for every n ∈ Z :
I1 ⊇ I2 ⊇ I3 ⊇ · · · ⊇ In ⊇ In+1 ⊇ · · ·
If lim (bn − an ) = 0, that is, if the lengths of the intervals In goes to 0 as n → ∞, then there is
n→∞
exactly one point L in the intersection
∞
\
In , that is, there is exactly one real number L such
n=1
∞
that L ∈ In for all n ∈ Z+ , and the sequences (an )∞
n=1 and (bn )n=1 both converge to L.
Bolzano-Weierstrass Theorem
Theorem 8. Every bounded sequence of real numbers has a convergent subsequence, that is,
if (an )∞
n=1 is a bounded sequence of real numbers, then there exists a convergent subsequence
∞
(ank )∞
k=1 of (an )n=1 .
Cauchy sequences
A sequence (an )∞
n=1 of real numbers is said to be a Cauchy sequence if for every real number
ε > 0, there exists N ∈ Z+ such that for all n ∈ Z+ and m ∈ Z+ ,
n>N
and
m>N
=⇒
|an − am | < ε.
The completeness property of R indeed means the following result:
Theorem 9. A sequence (an )∞
n=1 of real numbers is a Cauchy sequence if and only if it is
convergent.
References
[1] Course textbook: Joel Hass, Maurice D. Weir, and George B. Thomas, Jr. University
Calculus, Early Transcendentals, International Edition, 2nd edition, Pearson, 2012.
[2] Spivak, Michael. Calculus. Corrected 3rd edition. Cambridge University Press, 2006.
[3] Wade, William R. Introduction to Analysis. 4th edition, Pearson, 2010.
[4] Fitzpatrick, P. M. Advanced Calculus. Second Revised edition. American Mathematical
Society, 2009.
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