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 :
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:
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
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:
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.
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".
The Fibonacci sequence has been discovered already in 1202 by Leonardo Fibonacci . He investigated populations of rabbits, which approximately spread by the following rule:
At first, there is one pair of rabbits being able to mate.
A pair of rabbits being able to mate gives birth to another pair of rabbits every month.
A newborn pair takes one month where it cannot give birth to rabbits until it is finally able to do so.
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:
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 .
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:
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:
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.
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.
A sequence is bounded, if it has both an upper and a lower bound.
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
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?
alternating harmonic sequence
constant sequence: Bounded.
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.
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 .
harmonic sequence: The harmonic sequence is bounded from above by and from below by .
alternating harmonic sequence: This sequence is also bounded. An upper bound is and a lower bound is for .
Fibonacci sequence: Bounded from below (by 0), unbounded from above.
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?
alternating harmonic sequence
constant sequence: Both monotonously increasing and decreasing, but not strictly.
arithmetic sequence: For the sequence is strictly monotonously increasing. For the sequence is strictly monotonously decreasing. For we have a constant sequence.
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.
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".
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