MATH1051 Chapter 4: Chapter 4 - Sequences
Chapter 4 - Sequences
A sequence is a list of objects (usually numbers) with a definite order:
4.1 Formal Definition: Sequence
More formally, a sequence is a function, where the domain is (the set of natural
numbers, plus zero). Alternatively, you could simply take the domain as and start the
sequence at instead.
4.2 Representations
There are two main ways to represent terms in a sequence.
Direct:
Indirect:
The indirect method means you determine a term in a sequence by applying a function to
other elements in the range ( , not an element in the domain . In fact, you
could do this with as many elements in the range as you like, for example:
In order to define any term in an indirectly defined sequence, then, you need a starting
point. You’ll usually be given if you only need one term in the sequence, and and
if you need two.
4.3 Rabbit Population (Fibonacci Sequence)
There’s a very common example of this – the Fibonacci sequence. Here, each term is the sum
of the previous two numbers:
You’re given:
Then:
And you can go from there. We can apply this to an example – if we suppose that rabbits live
forever, and that every month each pair of rabbits produces a new pair which starts
reproducing when they’re two months old, and assume we have one newborn pair to begin
with… we can use that sequence to determine how many pairs there will be in the
month!
4.4 Limits
If we let the following be a sequence going from 0 to infinity:
Then:
This means that as gets larger and larger, gets arbitrarily close to . Exactly the
same as previous limits, except this is only defined for natural numbers . If the sequence
has a limit, it is convergent. If it isn’t, it’s divergent (though this doesn’t necessarily mean
the limit diverges off to infinity – it could simply oscillate and never approach one specific
value, like with a sin function).
So, if you’re given a function that determines , simply take the limit of and
you’ll find out what your term will end up being!
4.5 Theorem: Limit Laws
Exactly the same as before – don’t get stressed that there’s anything new to learn here.
These are not the new theorems you’re looking for. Set:
and . Then:
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Document Summary
A sequence is a list of objects (usually numbers) with a definite order: More formally, a sequence is a function, where the domain is numbers, plus zero). Alternatively, you could simply take the domain as sequence at instead. (the set of natural and start the. There are two main ways to represent terms in a sequence. The indirect method means you determine a term in a sequence by applying a function to other elements in the range ( could do this with as many elements in the range as you like, for example: In fact, you and if you need two. if you only need one term in the sequence, and. In order to define any term in an indirectly defined sequence, then, you need a starting point. There"s a very common example of this the fibonacci sequence. Then: forever, and that every month each pair of rabbits produces a new pair which starts.
