Mathematics: Sequences

Sequences
A sequence is a set of numbers that continues on in a defined pattern. The function that defines a sequence has a domain composed of the set of positive integers. Each member of the sequence is an element, or individual term. Each element is identified by the notation
is the term of the sequence, and <i>n</i> is the integer identifying which term in the sequence a is. There are two different ways to represent a sequence that contains the element ....;. The first is the simple notation . The expanded notation of a sequence is  . Notice that the expanded form does not end with the nth term. There is no indication that the nth term is the last term in the sequence, only that the nth term is an element of the sequence.

Arithmetic Sequences
An arithmetic sequence, or arithmetic progression, is a special kind of sequence in which each term has a specific quantity, called the common difference, that is added to the previous term. The common difference may be positive or negative. The general form of an arithmetic sequence containing n terms is
is the common difference. The formula for the general term of an arithmetic sequence is <br><img data-cke-saved-src=" />, where
 is the term you are looking for and d is the common difference. To find the sum of the first n terms of an arithmetic sequence, use the formula
.

Monotonic Sequences
A monotonic sequence is a sequence that is either nonincreasing or nondecreasing. A nonincreasing sequence is one whose terms either get progressively smaller in value or remain the same - a sequence that is bounded above. This means that all elements of the sequence must be less than a given real number. A nondecreasing sequence is one whose terms either get progressively larger in value or remain the same - a sequence that is bounded below. This means that all elements of the sequence must be greater than a given real number.

Recursive Sequences
When one element of a sequence is defined in terms of a previous element or elements of the sequence, the sequence is a recursive sequence.

For example, given the recursive definition
;
;
 for all
, you get the sequence 1, 1, 2, 3, 5, 8, … .

This particular sequence is known as the Fibonacci sequence, and is defined as the numbers zero and one, and a continuing sequence of numbers, with each number in the sequence equal to the sum of the two previous numbers. It is important to note that the Fibonacci sequence can also be defined as the first two terms being equal to one, with the remaining terms equal to the sum of the previous two terms. Both definitions are considered correct in mathematics. Make sure you know which definition you are working with when dealing with Fibonacci numbers.

Sometimes one term of a sequence with a recursive definition can be found without knowing the previous terms of the sequence. This case is known as a closed-form expression for a recursive definition. In this case, an alternate formula will apply to the sequence to generate the same sequence of numbers. However, not all sequences based on recursive definitions will have a closed-form expression. Some sequences will require the use of the recursive definition.

Golden Ratio and Fibonacci Sequence
The golden ratio is approximately 1.6180339887498948482… and is often represented by the Greek letter phi, Φ. The exact value of Φ is
 and it is one of the solutions to
.

The golden ratio is represented within the Fibonacci sequence, since the ratio of a term to the previous term approaches Φ as the sequence approaches infinity:




Geometric Sequences
A geometric sequence, or geometric progression, is a special kind of sequence in which each term has a specific quantity, called the common ratio, multiplied by the previous term. The common ratio may be positive or negative. The general form of a geometric sequence containing n terms is
is the common ratio. The formula for the general term of a geometric sequence is <br><img data-cke-saved-src=" />, where
 is the term you are looking for and r is the common ratio. To find the sum of the first n terms of a geometric sequence, use the formula
.

Any function with the set of all natural numbers as the domain is also called a sequence. An element of a sequence is denoted by the symbol
, which represents the nth element of sequence a. Sequences may be arithmetic or geometric, and may be defined by a recursive definition, closed-form expression or both. Arithmetic and geometric sequences both have recursive definitions based on the first term of the sequence, as well as both having formulas to find the sum of the first n terms in the sequence, assuming you know what the first term is. The sum of all the terms in a sequence is called a series. Consider the following example of a geometric sequence: Andy opens a savings account with $10. During each subsequent week, he plans to double the amount deposited during the previous week.

Sequence: 10, 20, 40, 80, 160, …
Function:

The sequence is a geometric sequence, with a common ratio of 2. All geometric sequences represent exponential functions. The nth term in any geometric sequence is represented by the general form,
, where
 represents the value of the nth term,
 represents the value of the initial term, r represents the common ratio, and n represents the number of terms. Thus, substituting the initial value of 10 and common ratio of 2 gives the function,
.

Limit of a Sequence
Some sequences will have a limit, or a value the sequence approaches or sometimes even reaches but never passes. A sequence that has a limit is known as a convergent sequence because all the values of the sequence seemingly converge at that point. Sequences that do not converge at a particular limit are divergent sequences. The easiest way to determine whether a sequence converges or diverges is to find the limit of the sequence. If the limit is a real number, the sequence is a convergent sequence. If the limit is infinity, the sequence is a divergent sequence.

Remember the following rules for finding limits:
· 
, for all real numbers k
· 

· 

· 
, for all real numbers k and positive rational numbers p
· The limit of the sums of two sequences is equal to the sum of the limits of the two sequences:
.

The limit of the difference between two sequences is equal to the difference between the limits of the two sequences:


The limit of the product of two sequences is equal to the product of the limits of the two sequences:


The limit of the quotient of two sequences is equal to the quotient of the limits of the two sequences, with some exceptions:
 . In the quotient formula, it is important to consider that
 and
.

The limit of a sequence multiplied by a scalar is equal to the scalar multiplied by the limit of the sequence:
is any real number.<br> <br> Infinite Series A. infinite series, also referred to as just a series, is a series of partial sums of a defined sequence. Each infinite sequence represents an infinite series according to the equation <br><img data-cke-saved-src=" />. This notation can be shortened to
 or
. Every series is a sequence of partial sums, where the first partial sum is equal to the first element of the series, the second partial sum is equal to the sum of the first two elements of the series, and the nth partial sum is equal to the sum of the first n elements of the series.

Every infinite sequence of partial sums (infinite series) either converges or diverges. Like the test for convergence in a sequence, finding the limit of the sequence of partial sums will indicate whether it is a converging series or a diverging series. If there exists a real number
 such that
, where
 is the sequence of partial sums, then the series converges. If the limit equals infinity, then the series diverges.

If   and
 is a real number, then
 is also the convergence value of the series.

To find the sum as n approaches infinity for the sum of two convergent series, find the sum as n approaches infinity for each individual series and add the results.
approaches infinity for the difference between two convergent series, find the sum as n approaches infinity for each individual series and subtract the results.


To find the sum as n approaches infinity for the product of a constant, also called a scalar, and a convergent series, find the sum as n approaches infinity for the series and multiply the result by the scalar.


The nth term test for divergence involves taking the limit of the nth term of a sequence and determining whether or not the limit is equal to zero. If the limit of the nth term is not equal to zero, then the series is a diverging series. This test only works to prove divergence, however. If the nth term is equal to zero, the test is inconclusive.

Problems:
P1. Suppose Rachel has $4,500 in her account in month 1. With each passing month, her account is one-half of what it was during the previous month. What would be a formula for the value of her account in any future month?

P2. Determine if the following geometric sequences converge or diverge:
(a)

(b)

(c)

(d)


P3. Determine a recursive expression for the following sequences:
(a)

(b)



P1. The sequence: 4500, 2250, 1125, 562.50, 281.25, … is geometric, since there is a common ratio of . Thus, this sequence represents an exponential function.

All geometric sequences represent exponential functions. Recall the general form of a geometric sequence is
, and substituting the initial value of 4500 and common ratio of
 gives
.

P2. (a)
. This sequence diverges.
(b)
. This sequence converges.
(c)
.

Since -1 raised to an integer power is either -1 or +1 this sequence just oscilates and does not converge. However, it is bounded above and below, so it does not diverge either.
(d)
. This

P3. (a) Use a table to determine the pattern:



Since the difference between successive terms is increasing at a uniform rate, we can use that to the guess at a sequence:
. Substitute the first terms to find d:



Thus, the recursive form would be

Use a table to determine the pattern:






that to the guess at a sequence:
. Substitute



Thus, the recursive form would be