Chapter 8: Infinite Sequences and Series
Section 8.1: Sequences
Table 8.1.1 summarizes some of the terms and issues that arise in the study of infinite sequences.
An infinite sequence is an ordered list of real numbers.
A formal definition: A function from the integers to the real numbers
Notation for a Sequence
ann=n0∞, where an is the general term for the sequence, and n0 is the starting index.
The notation is often shortened to an to save printing costs.
A sequence for which an+1≥an for all n≥n0
A sequence for which an+1≤an for all n≥n0
A sequence that is either increasing or decreasing
Sequence Bounded Above
A sequence for which an≤M for all n≥n0 and some (finite) real number M
Sequence Bounded Below
A sequence for which m≤an for all n≥n0 and some (finite) real number m
A sequence that is both bounded above and below
Limit of a Sequence
Informally: The numbers an approach the number L as n→∞.
Formally: The sequence an has limit L, that is, limn→∞an=L, if for each ϵ>0 there is an integer N for which an−L<ϵ for all n>N.
If limn→∞an=L, where L is a finite real number, then the sequence an is said to converge.
If limn→∞an=∞ or does not exist (because of oscillation), then the sequence an is said to diverge.
Table 8.1.1 Key terms related to infinite sequences
Table 8.1.2 lists two useful limits for sequences. If the index were a continuous variable, these limits could be obtained with the tools developed in Chapter 1.
limn→∞1nr=0 for r>0
Table 8.1.2 Two useful limits
Theorem 8.1.1 states that if f→L as the continuous variable x becomes infinite, then certainly the values of f at the integers must also approach L. Theorem 8.1.1 applies to those sequences whose general term an is the value of some function f at the integer n. In other words, Theorem 8.1.1 makes a statement about the limit of the sequence fnn=n0∞. If fx→L, then fn→L also.
If limx→∞fx=L, and an=fn, then limn→∞an=limn→∞fn=L.
Hence, Theorem 8.1.1 permits the application of the limit properties in Table 1.3.1, and any of the theory of indeterminate forms in Section 3.9 to be applied to sequences whose general term can be interpreted as the value of a function. In fact, Theorem 8.1.1 even permits the application of L'Hôpital's rule for such sequences.
A bounded monotone sequence converges.
Intuitively, it is appealing that a sequence that is either increasing or decreasing, and that is also bounded, should converge. However, Theorem 8.1.2 is really a deep statement about the real numbers: it declares that there is actually a real number L to serve as the limit of the sequence. Hence, this is as much a statement about the completeness of the real numbers as it is about the behavior of certain sequences.
Theorem 8.1.3: The Discrete Squeeze Theorem
an≤bn≤cn for all n>N for some N
an→L and cn→L as n→∞
Theorem 8.1.3 is the discrete analog of the Squeeze theorem stated for continuous functions in Section 1.3.
If an=2 n2−3 n4 n3+5, find the limit of the sequence ann=0∞.
If an=lnnn, find the limit of the sequence ann=1∞.
If bn=−2−nn, find the limit of the sequence bnn=1∞.
If an=n!nn, find the limit of the sequence ann=1∞.
If an=nn2+1, show the sequence ann=0∞ is decreasing.
If an=5nn!, find the limit of the sequence ann=0∞.
If an=1⋅3⋅5⋅⋯⋅2 n−1nn, show that the sequence ann=1∞ is decreasing.
If a1=1,a2=−1, and 6 an+2−5 an+1+an=0 defines an for n>2, use Maple to find the general term an.
If a1=1 and an+1=43+2 an for n>1, graph an for n=1,…,10.
Use Maple to find an explicit representation for an.
Use Maple to calculate limn→∞an.
If an=5 n−42 n+3, determine if the sequence ann=0∞ converges or diverges.
If it converges, find the limit of the sequence.
If an=n3+n4n+n5, determine if the sequence ann=1∞ converges or diverges.
If an=n!n+2!, determine if the sequence ann=0∞ converges or diverges.
If an=n+5−n, determine if the sequence ann=0∞ converges or diverges.
If an=n!3n, determine if the sequence ann=0∞ converges or diverges.
If an=n1/n, determine if the sequence ann=1∞ converges or diverges.
If an=2⁢n+5n2+7⁢n+2, determine the limit of the sequence ann=0∞.
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