Examples and properties of sequences – Serlo

ExamplesBearbeiten

Constant sequenceBearbeiten

 
Example for a constant sequence:   for all  

A sequence is called constant, if all of its elements are equal. An example is:

 

With   , the general form of a constant sequence is   for all  .

Arithmetic sequencesBearbeiten

 
Example for an arithmetic sequence:   for all  

Arithmetic sequences have constant differences between two elements. For instance, the sequence of odd numbers is arithmetic, since any two neighbouring elements have difference  :

 

A further example is sequence   with   for all  :

 

Question: What is the recursive rule for a general arithmetic sequence?

The first element   may be imposed arbitrarily. The next one has any constant distance from  .Let's call this difference  . Then,   and hence   . Analogously, since   there is   and so on. So we have the recursive definition:

 

Question: What is the explicit rule for a general arithmetic sequence?

We already know the recursive rule   for all  , where   Is given. That means   and  . Analogously  . So we get an explicit rule for all  :

 

Geometric sequenceBearbeiten

 
Example for a geometric sequence:   for all  

For the geometric sequence we have a constant ratio between two subsequent elements. No element is allowed to be 0, since else we would get into trouble dividing by 0 when computing ratios. An example for a geometric sequence is   where the constant ratio is given by  :

 

Question: What is the explicit rule for a general geometric sequence?

The first element   for a geometric series is arbitrary. The second element must have a fixed ratio to  . Let us call this ratio  . This means  , or equivalently   . Now, as the ratio is always fixed, all other elements are given at this point. We have   or equivalently   and so on. Hence, the recursive definition reads:

 

Question: What is the explicit rule for a general geometric sequence?

Let us take the recursive rule   and try to find an explicit formulation. There is   and  . Analogously, . This suggests:

 

which can easily be checked by induction.

Harmonic sequenceBearbeiten

 
The first ten element of the harmonic sequence

The sequence   is called harmonic sequence. The name originates from the fact that intervals in music theory can be defined by it: It describes octaves, fifths and thirds. Mathematicians like it, because it is one of the smallest sequences where the sum over all elements gives infinity (we will com to this later, when concerning series).The first elements of this sequence are:

 

The similar sequence   or   is called alternating harmonic sequence . Explicitly, the first elements are

 

or

 

For   the generalized harmonic sequence is given by

 

Alternating sequencesBearbeiten

 
Example for an alternating sequence:   for all  

An alternating sequence is characterized yb a change of sign between any two sequence elements. The term "alternating" just means that the presign is "constantly changing". For instance, the sequence   alternates between the values   and  , so we have an alternating sequence . A further example is   with  .

More generalyl, any alternating sequence can be put into the form:

  1.  
  2.  

Here,   is a sequence of non-negative numbers.

Question: Which alternating sequences can be brought into which of the above two forms?

This is answered by taking a look at the first sequence element. If the first element with index 0 is positive( ), then we have the form  . Conversely, if   the we have  .

If the first index is 1 (the sequence starts with  , it is exactly the other way round.

If the first element is   , one needs to check the subsequent elements until one is strictly greater or smaller than 0. Only the sequence which is constantly 0 takes both forms at once.

The exponential sequenceBearbeiten

To-Do:

Sketch the first sequence elements

A common example for a sequence is the exponential sequence. For instance, it appears when you invest money and get a return (e.g. in terms of interests). For instance, imagine you invest one "money" of any currency (dollar or pound or whatever) at a bank with a rate of interest of   (oh my gosh, what a bank!) Then, after one year, you will get paid back   "moneys" (2 units of money). Is there a way to get more money, if you are allowed to spread   of interests over a year? You could ask the bank to pay you an interest rate of  , but twice a year. Then, after one year where multiplying your money twice, you get back

 

units of money. Those are   units more! If you split the interest rate in even smaller parts, you get even more: for 4 times  , you get back   units of money.

Question: Why do you get back   units if you split the interest rate of   in 4 equal parts of  ?

After the first 3 months, the amount of money you have will be   . Another 3 months later, it will have increased by the same factor and you get   (approximately). Doing this two more times, you get

 

units of money.

In general, if you split the   into   parts, then in the end you will receive

 

units of money. This can be interpreted as a sequence in  : the sequence   is also called "exponential sequence". Now, can you make infinitely much money within one year, just by splitting the   infinitely often? The answer is: unfortunately no. There is an upper bound to how much money you can make that way. It is called Euler's number  . So you do not get above   units of money. The proof why this sequence   converges to   can be found within the article "monotony criterion".

Sequence of Fibonacci numbersBearbeiten

 
The first elements of the Fibonacci sequence
 
How to determine the Fibonacci sequence

The Fibonacci sequence has been discovered already in 1202 by Leonardo Fibonacci . He investigated populations of rabbits, which approximately spread by the following rule:

  1. At first, there is one pair of rabbits being able to mate.
  2. A pair of rabbits being able to mate gives birth to another pair of rabbits every month.
  3. A newborn pair takes one month where it cannot give birth to rabbits until it is finally able to do so.
  4. We consider an ideal world, with no rabbits leaving, no predators, infinitely much food and no rabbits dying.

Question: How many rabbits will be there in each month?

Let   be the number of rabbit pairs being able to mate within month  . What is then   ? Within this month, an additional amount of rabbit pairs   will become able to mate. So there is  . But now, the newly mating rabbit pairs in month   are exactly those born in month   (because rabbits are born, then they take a pause of one turn and after 2 months, they start to mate). I.e.  . Plugged into the above equation, we get:

 

Or after an index shift:

 

In the beginning, there is   and   (rabbits born in month 1 only start to mate in month 3). So we have a recursively defined sequence, where each   can be determined if   and   are known. This sequence is also called Fibonacci sequence. The shorthand definition reads:

 

Its first elements are  

Mixed sequencesBearbeiten

Mixed sequences are a generalization of alternating sequence. We merge two sequences   and   into a new one which consists alternately of elements of   and  , i.e.

 

An element with odd index, e.g.   for   will be equal to   from the sequence   . And an element with even index, e.g.   for   agrees with   from the sequence   .

In order to get a general formula for   with   , we just have to distinguish the cases of even and odd   . For odd   , there is   or equivalently   , so we get  . For an even   there is  . Together, we have

 

  is then said to be a mixed sequence composed by   and  .

Example (mixed sequence)

The alternating sequence   given by   ( ) is a merger of the sequences   and   , since for   there is

 

If you encounter an exercise where a sequence is defined with a distinction between even and odd   , then it is just a mixed sequence. Basically, any sequence can be interpreted as a mixed sequence: Any   is composed by   and  . For instance  can be seen as a merger of   and  .

Question: Are there sequences which remain invariant, if they are merged with itself?

Yes, but only constant sequences.

For   , if we mix the constant sequence   with itself  , we again get the constant sequence  .

Conversely, if   is a mixture by   and  , then

 

For any element   we can apply this formula. Since for   there is   or   , we get smaller and smaller indices until we reach   . So the sequence has to be constant with only value  .

Properties and important termsBearbeiten

Bounded sequenceBearbeiten

 
An example for a bounded sequence ( ) with some bounds.


a sequence is called bounded from above, if there is an upper bound, i.e. a large number, which is never exceeded by any sequence element. This number bounds the sequence from above. The mathematical definition of this expression reads:

 

Or explicitly:

 

Analogously, a sequence is bounded from below if and only if there is a lower bound, i.e. a number for which all sequence elements are greater than this number. The mathematical definition hence reads:

 

Question: What does it mean that a sequence in not bounded from below or above?

We can formally invert the above statements:

 

So a sequence is unbounded from above, if for any   there is some sequence element bigger than  . That means, parts of the sequence grow infinitely big. Conversely,:

 

So a sequence is unbounded from below, if for any   there is a sequence element smaller than  . Intuitively, a part of the sequence gets infinitely small.

If a sequence is both bounded from above and from below, we just call it bounded. So we have the following definitions:

upper bound
An upper bound is a number, which is greater than any sequence element. So   is an upper bound of  , if and only if   for all  .
sequence bounded from above
A sequence is bounded from above, if it has any upper bound.
lower bound
A lower bound is a number, which is smaller than any sequence element. So   is a lower bound of  , if and only if   for all  .
sequence bounded from below
A sequence is bounded from below, if it has any lower bound.
bounded sequence
A sequence is bounded, if it has both an upper and a lower bound.

Hint

An upper bound does not need to be the smallest (best) upper bound. And a lower bound does not need to be the greatest lower bound, either. For instance, if a sequence is bounded from above by   , then  ,  ,   and   are upper bounds, as well. Boundedness from above can be shown by just stating any upper bound.

There is an alternative definition of boundedness:

Theorem (alternative definition of boundedness)

A sequence is bounded if and only if there is a real number   , such that for all elements   there is   .

Proof (alternative definition of boundedness)

This is equivalent to the first definition: We can show that

 

or explicitly:

 

So we have to prove equivalence, which means that we have to prove both directions of the above double arrow. Let us start with the first direction: We assume that the first definition is fulfilled by a sequence. Thus, there are real numbers  , such that for all sequence elements   holds. Then for all sequence elements there is also  . So the existence of some   within the alternative definition is established (  can be any positive, real number greater than or equal to  ).

And what about the other direction? Let now be   given, with   for all sequence elements. Then the inequality   holds for all sequence elements. Thus   represents a lower bound and   an upper bound for the sequence  , so that the sequence is also bounded by the first definition.

Question: Which of the following sequences is bounded/unbounded from above/below?

  1. constant sequence
  2. arithmetic sequence
  3. geometric sequence
  4. harmonic sequence
  5. alternating harmonic sequence
  6. Fibonacci sequence

Lösung:

  1. constant sequence: Bounded.
  2. arithmetic sequence: For   it is bounded from below by   and unbounded from above. For   it is bounded from above by   and unbounded from below. For   we have a constant sequence, which is bounded.
  3. geometric sequence: For   we have a constant sequence, which is again bounded. For   and   the sequence is bounded from below (by  ) and unbounded from above. For   and   it is the other way round: The sequence is bounded from above by the negative number  but unbounded from below. For   the absolute values grow infinitely large and the sign is alternating. So the sequence is unbounded both from above and from below. If we choose   and   , it is bounded. The upper bound is   and the lower one  . For   and   the lower bound is   and the upper bound is  . So the sequence is bounded. And for   we also have boundedness by   and  .
  4. harmonic sequence: The harmonic sequence  is bounded from above by   and from below by  .
  5. alternating harmonic sequence: This sequence is also bounded. An upper bound is   and a lower bound is   for  .
  6. Fibonacci sequence: Bounded from below (by 0), unbounded from above.

Monotone sequencesBearbeiten

Sequences are also distinguished according to their growth behaviour: If the sequence elements of become larger and larger (i.e. each subsequent sequence member   is larger than  ), this sequence is called a strictly monotonically growing/increasing sequence. Similarly, a sequence with ever smaller sequence elements is called a strictly monotonously falling/decreasing sequence. If you want to allow a sequence to be constant between two sequence elements, the sequence is called only monotonously growing/increasing sequence or monotonously falling/decreasing sequence (without the "strictly"). Remember: "strictly monotonous" means as much as "getting bigger and bigger" or "getting smaller and smaller". In contrast, "monotonous", without the "strict", means as much as "getting bigger and bigger or remaining constant" or "getting smaller and smaller or remaining constant". The mathematical definition is:

Definition (monotone sequences)

For a real sequence   we define:

 

Question: Which of the following sequences are monotonously increasing/decreasing? For which ones, the monotony is strict?

  1. constant sequence
  2. arithmetic sequence
  3. geometric sequence
  4. harmonic sequence
  5. alternating harmonic sequence
  6. Fibonacci sequence

Lösung:

  1. constant sequence: Both monotonously increasing and decreasing, but not strictly.
  2. arithmetic sequence: For   the sequence is strictly monotonously increasing. For   the sequence is strictly monotonously decreasing. For   we have a constant sequence.
  3. geometric sequence: For   and   it is strictly monotonously increasing and for   it is strictly monotonously decreasing. For   and   it is strictly monotonously decreasing and for   strictly monotonously increasing. For   we have an alternating sequence, which is neither monotonously increasing, nor decreasing. For   the sequence is constant.
  4. harmonic sequence: strictly monotonously decreasing.
  5. alternating harmonic sequence: alternating and hence neither monotonously increasing, nor decreasing.
  6. Fibonacci sequence: monotonously increasing.

Remark: convergent sequencesBearbeiten

Sequences are also distinguished by whether they have a limit or not. Sequences which have a limit are called convergent and all other ones are divergent. This property requires a bit more explanation. We will come back to it later within the article "convergence and divergence".