Sequences
Videos
Review Videos
Videos, Math Dr. Bob:
Link to original
- Infinite sequences: Definition; Squeeze Theorem
- Extra: Infinite sequences: Various examples, arithmetic and geometric
- Extra: Infinite sequences: Recursive sequences (like Fibonacci)
01 Theory
Theory 1
A sequence is a rule that defines a term for each natural number
: So a sequence is a function from
to . Geometric sequence
A sequence is called geometric if the ratio of consecutive terms is some constant
, independent of : The defining relation of a geometric sequence is equivalent to
. By plugging
into , we have . This plugging can be repeated -times to get a formula for the term: Therefore
, and we have a formula for the general term of the sequence (the term with index ).
Starting point of a sequence
Note that sometimes the index (variable) of a sequence starts somewhere other than
. Most common is but any other starting point is allowed, even negative numbers. Sometimes
is used instead of in the formula for the general term of a sequence, thus . The ‘ ’ notation is useful when the sequence starts from . Link to originalExtra - Fibonacci sequence
The Fibonacci sequence goes like this:
The pattern is:
This formula is a recursion relation, which means that terms are defined using the values of prior terms.
The Fibonacci sequence is perhaps the most famous sequence of all time. It is related to the Golden Ratio and the Golden Spiral:
02 Illustration
Example: Geometric sequence - revealing the format
Geometric sequence: revealing the format
Find
and and (written in the geometric sequence format) for the following geometric sequences: (a)
(b)
(c)Solution
(a) Plug in
to obtain . Notice that and so therefore . Then the ‘general term’ is .
(b) Rewrite the fraction:
Plug that in and observe
. From this format we can read off and .
(c) Rewrite:
From this format we can read off
Link to originaland .
Series
Videos
Review Videos
Videos, Math Dr. Bob:
Link to original
- Infinite series: Definitions, basic examples
- Geometric series and SDT: Geometric series, Simple Divergence Test (aka “Limit Test”)
- Infinite series: Various examples
- Extra: Infinite series convergence: Telescoping series
03 Theory
Theory 1
A series is an infinite sum that is created by successive additions without end. The terms are not added up “all at once” but rather they are added up “as
increases” or “as .” Three of the most famous series are the Leibniz series and the geometric series:
Link to originalPartial sum sequence of a series
The partial sum sequence of a series is the sequence whose terms are the sums up to the given index:
These
terms themselves form a sequence:
04 Illustration
Example: Geometric series - total sum and partial sums
Geometric series - total sum and partial sums
The geometric series total sum
can be calculated using a “shift technique” as follows: (1) Compare and : (2) Subtract second line from first line, many cancellations:
(3) Solve to find
: Note: this calculation assumes that
exists, i.e. that the series converges. The geometric series partial sums can be calculated similarly, as follows:
(1) Compare
and : (2) Subtract second line from first line, many cancellations:
(3) Solve to find
: (4) The last formula is revealing in its own way. Here is what it means in terms of terms:
Link to original
Convergence
Videos
Review Videos
Videos, Math Dr. Bob:
Link to original
- Infinite sequences convergence: Squeeze; Monotone Bounded
- Infinite sequences convergence: Examples sequences: convergent, monotonic, bounded
05 Theory
Theory 1
A sequence has a limit if its terms tend toward a specific number, or toward
.
When this happens we can write “
” with some number or . We can also write “ as ”. The sequence is said to converge if it has a finite limit
. Some sequences don’t have a limit at all, like
:
Or
:
These sequences diverge. In the second case, there is a limit
, so we say it diverges to . A sequence may have a limit of
but is still said to diverge. Extra - Convergence definition
The precise meaning of convergence is this. We have
as if, given any proposed error , it is possible to find such that for all we have . When
, convergence means that given any , we can find such that for all we have . Similarly for
.
If the general term
is a continuous function of , we can replace with the continuous variable and compute the continuous limit instead: If
Link to originalwould be a differentiable function, and we discover an indeterminate form, then we can apply L’Hopital’s Rule to find the limit value. For example, if the indeterminate form is , we can convert it to and apply L’Hopital.
06 Illustration
Example - L’Hopital’s Rule for sequence limits
L’Hopital’s Rule for sequence limits
(a) What is the limit of
? (b) What is the limit of ? (c) What is the limit of ? Solution (a) Identify indeterminate form
. Change from to and apply L’Hopital:
(b) Identify indeterminate form
. Change from to and apply L’Hopital:
(c) (1) Identify form
and rewrite as : (2) Change from
to and apply L’Hopital: (3) Simplify:
(4) Consider the limit:
Link to original
Example - Squeeze theorem
Squeeze theorem
Use the squeeze theorem to show that
as . Solution
(1) We will squeeze the given general term above
and below a sequence that we must devise: (2) We need
to satisfy and . Let us study . (3) Now for the trick. Collect factors in the middle bunch:
(4) Each factor in the middle bunch is
so the entire middle bunch is . Therefore: Now we can easily see that
Link to originalas , so we set and we are done.