Elementary Algebra and Calculus: The Whys and Hows by Fradkin Larissa

Author:Fradkin, Larissa [Fradkin, Larissa]
Language: eng
Format: epub
Publisher: UNKNOWN
Published: 2021-05-03T16:00:00+00:00

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13.1.4 Defining a Sequence via a Recurrence Relationship

A sequence element may be defined in another way, via a recurrence relation

xn + 1 = f(xn , xn–1 , ..., x1 ),

where f may be a function of several arguments and not just one argument,

Thus, there are two ways to describe a sequence,

1. using a functional relation x(n), which specifies how each sequence element is defined by its counter;

2. using a recurrence relation xn + 1 = f(xn , xn–1 , …, x1 ), which specifies how each sequence element is defined by previous sequence element(s).

Examples:

1. Given a sequence xn = n2 we can change the functional description to a recurrence relation

xn + 1 = (n + 1)2 = n2 + 2n + 1

⇒ xn + 1 = xn + 2 √xn + 1

When given such a recurrence the first element needs to be specified. Only then can we start evaluating other elements.

2. A Fibonacci sequence : 1, 1, 2, 3, 5, 8, 13, 21, … can be described via a recurrence relation

xn + 1 = xn + xn -1 . When given such a recurrence the first two elements have to be specified. Only then can we start evaluating other elements. Let us check that the above recurrence describes the given sequence:

Question: What are x1 and x2 ?

Answer:

Question: Does x3 satisfy the given recurrence relation and why? Answer:

Question: Does x4 satisfy the given recurrence relation and why? Answer:

Question: Does x5 satisfy the given recurrence relation and why? Answer:

Fibonacci sequences in nature

When superimposed over the image of a nautilus shell we can see a Fibonacci sequence in nature:

http://munmathinnature.blogspot.com/2007/03/fibonacci-numbers.html

Each of the small spirals of broccoli below follows the Fibonacci’s sequence.

http://www.pdphoto.org/PictureDetail.php?mat=pdef&pg=8232

13.2 Limit of a sequence

Taking a limit of a sequence as n grows larger and larger without bounds is the first advanced operation on functions that we cover. In mathematics, instead of the phrase n grows larger and larger without bounds we use the shorthand n ∞ (verbalised as n tends to infinity ). If it exists the outcome of applying this operation to a (discrete) function x n is called lim x n and is either a number or else ± ∞ (either +infinity or infinity ). Sometimes instead of lim x n we write lim x n . However, usually, the condition n → ∞ is understood and not mentioned. n →∞

Note: ∞ is not a number but a symbol of a specific sequence behavior, ∞ means that the sequence increases without bounds and ∞ means that the sequence decreases without bounds – see the right column in the Table in Section 13.2.1 below.

13.2.1 Definition of a

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