Infinite Sequences

Sequences Recap

• A sequence is any ordered list of numbers.
  • They can be expressed in the form:

• Each number in a sequence is a term.

• We often use as the index variable for the sequence.

  • The index variable allows us to refer to a specific term in a sequence.
    (Example: is the term in the sequence; is the second term in the sequence.)

• An infinite sequence is a sequence with an infinite number of terms.

• We use the notation below to denote an infinite sequence

Defining Sequences

• Consider the sequence below:

• You might identify that there is a pattern within each consecutive term in a sequence.

• If we assume that this pattern continues, we can define the next terms of this sequence as an explicit formula.

• Therefore, we can refer to this sequence as:

• We can also define this sequence recursively.
  • We first define the first term in our sequence . (In our case, )
    • This is called the base case.
  • Then, we define the rule on how to find the next term in our sequence.
    • Since each consecutive term is achieved b multiplying to the preceding term, we can now define our sequence recursively:

Special Sequences

• Some types of sequences occur more often and are given special names.
• These are the arithmetic sequences and geometric sequences.

Arithmetic Sequences

• Consider the following sequence:

• Notice that we get the next term by adding to a preceding term.
  • This is because the difference between each two consecutive terms is constant.
  • Therefore, we can describe this sequence via the recurrence relation.

• Knowing that:

• We can express the same sequence using the explicit formula.

Definition
Arithmetic Sequence

An arithmetic sequence is a sequence where the difference of each two consecutive terms is a constant.

It has the general explicit form:

where:

• term of the sequence
• first term of the sequence
• common difference

It also has the general recursive form:

where:

• succeeding term of the sequence
• first term of the sequence
• preceding term of the sequence
• common difference

Geometric Sequences

• Consider the following sequence.

• This time, we get the succeeding term by multiplying from the previous term.
  • Therefore, we can express it recursively as:

• We can also express this sequence using an explicit relation.
• Since we know that:

• Then we can express the same sequence as an explicit function

Definition
Geometric Sequence

A geometric sequence is a sequence where the ratio of each two consecutive terms is a constant.

It has a general explicit form of:

where:

• term of the sequence
• first term of the sequence
• common ratio

It also has the general recursive form:

where:

• suceeding term of the sequence
• first term of the sequence
• preceding term of the sequence
• common ratio

Limit of a Sequence

• When we consider infinite sequences, we usually ask what happens to the behavior of the terms in an infinite sequence as gets larger.

  • We can prove using some examples that some sequences produce interesting behavior as the number of terms get larger.

• Example 1:

  • Here, as , the sequence as well.

• Example 2:

  • As , the sequence seem to approach .

• Example 3:

  • As , the terms alternate between and but they do not approach a single value

• Example 4:

  • As , the terms alternate but .

• In some sequences, the terms seem to approach some finite value as .

  • If they do, then we say that the limit of a sequence as is some finite number .

Definition
Limit of a Sequence

Given a sequence , if the terms become arbitrarily close to some finite number as becomes sufficiently large, then we say that is the limit of the sequence

• With this, the existence of a limit of a sequence allows us to define the convergence or divergence of a sequence.

Definition
Convergence or Divergence of a Sequence

Given a sequence , if there is a real number such that:

then is a convergent sequence.

If a sequence does not converge, then it is a divergent sequence and we say that the limit does not exist.

Note

• When a sequence diverges, it means that rather than it approaches a finite value as , it instead either goes unbounded, or it doesn’t approach a single value.
  • In such cases that the sequence goes unbounded, we use the notation or .
  • Take note that the limit still doesn’t exist and we use such notation to denote how the sequence diverges.

Evaluating Limits of Sequences

• When a sequence is defined by an explicit function , we can simply use properties of limits of functions to determine the convergence of a sequence.

Theorem
Limit of a Sequence Defined by a Function

Given a sequence defined by the explicit function for all :

If there is a real number such that:

then converges and

• We can use this theorem to evaluate the limit of a geometric sequence.

Proof
The Limit of a Geometric Sequence for

Consider the geometric sequence and the related exponential function

• If : ^lim-geo-seq-converge-poscase
  • Then,
  • Therefore the sequence converges and its limit is also .
• If :
  • Then,
  • Therefore, the sequence also converges and its limit is also .
• If :
  • Then,
  • Therefore, this theorem cannot be applied.
  • However, in this case, since as , the terms of the sequence also approaches as .
  • Therefore, the sequence diverges for .

Therefore, to summarize:

Definition
The Limit of a Geometric Sequence

Given a geometric sequence defined by the explicit function with a common ratio , its limit is:

• Some sequences may be defined as an algebraic combination of two sequences and .
  • In this case, we use the algebraic limit laws to evaluate such sequences.

Definition
Algebraic Limit Laws

Given two sequences and and any real number :
If there exist constants and such that

then these following properties should apply:

• Some other limit properties for functions can be extended to sequences as well.
  • For example, we can extend the notion of the composite function theorem and the squeeze theorem to apply for limits of sequences as well.

Theorem
Limit of a Composition of a Function and a Sequence

Let be a real number such that the sequence converges to .

Suppose that is continuous at . Then there is a minimum index such that is defined for all terms with an index . Then:

• This theorem says that if a sequence is inside a function , then its limit is the same as the value of at the limit of the sequence .
  • However, keep that in mind that for this to be true, must be continuous at the limit of .

Definition
Squeeze Theorem for Sequences

Consider the sequences , , and .
Suppose there exists a minimum index where:

If there exists a real number such that

then converges and

• With this, we can complete the limit of a geometric sequence for sequences with a negative common ratio.
  • This way, we can define the limit of a geometric sequence as we consider all cases with a negative
  • In addition, some strategies may be required when proving how some sequences that alternate diverge.

Proof
Limit of a General Geometric Sequence

Consider the geometric sequence and the related exponential function .

• If :
  • Then, see here.
• If : ^lim-geo-seq-negcase
  • Then, via the inverse property of multiplication, there exists a real number where and
  • Since ,
    • by proving that the sequences and both converge to the same value , then by squeeze theorem
  • As , .
  • Similarly, as ,
  • Therefore, via the squeeze theorem:
    • Substituting ,
• If :
  • Consider two sequences: and
    • represents the terms of sequence with an even index.
    • represents the terms of sequence with an odd index.
  • If both of these subsequences converge to the same value , then also converges to . Otherwise, the sequence diverges.
    • For :
      • Applying laws of exponents:
      • Therefore, and the sequence converges to .
    • For :
      • Applying laws of exponents:
      • Therefore, and the sequence converges to .
  • Since , then the sequence diverges
• If :
  • Consider two sequences: and
    • represents the terms of the sequence with an even index.
    • represents the terms of the sequence with an odd index.
  • If both of these subsequences converge to the same value , then also converges to . Otherwise, the sequence diverges.
    • For :
      • Applying laws of exponents: .
      • Since we know that is negative, must be positive.
      • By establishing that , we can deduce that this case is the same as when .
      • Therefore, as , then .
    • For :
      • Applying laws of exponents: .
      • Since we know that is negative and is positive, then must be negative.
      • Therefore, as , .
    • Since both of them diverges, then must diverge as well.
    • But since , then they diverge in both directions.

Therefore, the limit of a geometric sequence given a common ratio is given as:

Definition
Limit of a Geometric Sequence

Given a geometric sequence with a common ratio , its limit is defined as:

Bounded Sequences

• We start by defining the much needed terminology and motivation for an important theorem involving sequences: Monotone Convergence Theorem.
• First, we define what it means for a sequence to be bounded.

Definition
Bounded Sequences

• A sequence is bounded above if there exists a real number such that for all positive integers :

• A sequence is bounded below if there exists a real number such that for all positive integers :

• A sequence is a bounded sequence if it is bounded above and bounded below.
• If a sequence is not bounded, then it is an unbounded sequence.

• Here are some examples on how a sequence might be bounded.

  • Example 1:
    • Since for all positive integers , then this sequence is bounded above.
    • Since for all positive integers , then this sequence is bounded below.
    • Therefore, is a bounded sequence.
      
  • Example 2:
    • Since for all positive integers , then this sequence is bounded above
    • However, this sequence is not bounded below.
    • Therefore, is an unbounded sequence.

• There is a relationship between boundedness and convergence.

  • If a sequence is unbounded, then it is not bounded above, below, or both.
  • Then, there might be terms that might arbitrarily get large as gets larger.
  • Therefore, cannot converge.
  • This is why the boundedness of a sequence is a necessary condition for a sequence to converge.

Theorem
Boundedness and Convergence

If a sequence converges, then it is bounded.

• However, boundedness isn’t a sufficient condition for a sequence to converge.
  • For example, is bounded, but the sequence diverges as it never approaches a finite value.
• Therefore, an other condition must be necessary to ensure a bounded sequence must converge.

Definition
Monotony of a Sequence

• A sequence is increasing for all if

• A sequence is decreasing for all if

• A sequence is monotone for all if it is increasing or decreasing for all .

• What this means is that:

  • A sequence is increasing if the next term is greater than or equal to the previous term.
  • A sequence is decreasing if the next term is less than or equal to the previous term.
  • The condition is necessary to ensure that there is a minimum index where this property remains consistent.
    • This simply says that terms before may not be consistently increasing or decreasing, but will eventually increase or decrease after .

• We now have the necessary requirements to state the Monotone Convergence Theorem.

Theorem
Monotone Convergence Theorem

If is a bounded sequence and there exists a positive integer where is monotone for all , then converges.