Serlo: EN: Examples and properties of sequences

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Examples

Constant sequence

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

With $c\in ℝ$ , the general form of a constant sequence is $a_n:=c$ for all $n\in \mathbb{N}$.

Arithmetic sequences

Arithmetic sequences have constant differences between two elements. For instance, the sequence of odd numbers is arithmetic, since any two neighbouring elements have difference $2$:

Geometric sequence

Example for a geometric sequence: $a_n=2^n$ for all $n\in \mathbb{N}$

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 $a_n=2^n$ where the constant ratio is given by $2$:

which can easily be checked by induction.

Harmonic sequence

The first ten element of the harmonic sequence

The sequence $a_n=\frac{1}{n}$ 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 $a_n=(-1{)}^n\cdot \frac{1}{n}$ or $b_n=(-1{)}^{n+1}\cdot \frac{1}{n}$ is called alternating harmonic sequence . Explicitly, the first elements are

or

For $k\in \mathbb{N}$ the generalized harmonic sequence is given by

Alternating sequences

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 $a_n=(-1{)}^n$ alternates between the values $1$ and $-1$, so we have an alternating sequence . A further example is $a_n=(-1{)}^{n+1}\cdot n$ with ${\left(a_n\right)}_{n\in \mathbb{N}}=(1,-2,3,-4,5,-6,\dots )$.

The exponential sequence

Vorlage:Todo 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 $100\mathrm{\%}$ (oh my gosh, what a bank!) Then, after one year, you will get paid back $1+1=2$ "moneys" (2 units of money). Is there a way to get more money, if you are allowed to spread $100\mathrm{\%}$ of interests over a year? You could ask the bank to pay you an interest rate of $50\mathrm{\%}$, but twice a year. Then, after one year where multiplying your money twice, you get back

units of money. Those are $0.25$ units more! If you split the interest rate in even smaller parts, you get even more: for 4 times $25\mathrm{\%}$, you get back $2.44$ units of money.

In general, if you split the $100\mathrm{\%}$ into $n$ parts, then in the end you will receive

units of money. This can be interpreted as a sequence in $n$: the sequence ${\left({(1+\frac{1}{n})}^n\right)}_{n\in \mathbb{N}}$ is also called "exponential sequence". Now, can you make infinitely much money within one year, just by splitting the $100\mathrm{\%}$ 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 $e=2.71828\dots$. So you do not get above $2.72$ units of money. The proof why this sequence ${\left({(1+\frac{1}{n})}^n\right)}_{n\in \mathbb{N}}$ converges to $e$ can be found within the article "monotony criterion".

Sequence of Fibonacci numbers

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.

Mixed sequences

Mixed sequences are a generalization of alternating sequence. We merge two sequences $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$ into a new one which consists alternately of elements of $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$, i.e.

An element with odd index, e.g. $a_{2k-1}$ for $k\in \mathbb{N}$ will be equal to $b_k$ from the sequence $(b_n{)}_{n\in \mathbb{N}}$ . And an element with even index, e.g. $a_{2k}$ for $k\in \mathbb{N}$ agrees with $c_k$ from the sequence $(c_n{)}_{n\in \mathbb{N}}$ .

In order to get a general formula for $a_n$ with $n\in \mathbb{N}$ , we just have to distinguish the cases of even and odd $n$ . For odd $n$ , there is $k=\frac{n+1}{2}$ or equivalently $n=2k-1$ , so we get $a_n=a_{2k-1}=b_k=b_{\frac{n+1}{2}}$. For an even $n$ there is $a_n=c_{\frac{n}{2}}$. Together, we have

$(a_n{)}_{n\in \mathbb{N}}$ is then said to be a mixed sequence composed by $(b_n{)}_{n\in \mathbb{N}}$ and $(c_n{)}_{n\in \mathbb{N}}$.

If you encounter an exercise where a sequence is defined with a distinction between even and odd $n$ , then it is just a mixed sequence. Basically, any sequence can be interpreted as a mixed sequence: Any $(a_n{)}_{n\in \mathbb{N}}$ is composed by $(a_{2n-1}{)}_{n\in \mathbb{N}}$ and $(a_{2n}{)}_{n\in \mathbb{N}}$. For instance $(1,2,3,\dots )$can be seen as a merger of $(1,3,5,\dots )$ and $(2,4,6,\dots )$.

Properties and important terms

Bounded sequence

An example for a bounded sequence ($a_n=(-1{)}^{n+1}\cdot \frac{1}{n}$) 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:

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 $S$ is an upper bound of $(a_n{)}_{n\in \mathbb{N}}$, if and only if $a_n\le S$ for all $n\in \mathbb{N}$.

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 $S$ is a lower bound of $(a_n{)}_{n\in \mathbb{N}}$, if and only if $a_n\ge S$ for all $n\in \mathbb{N}$.

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.

Monotone sequences

Sequences are also distinguished according to their growth behaviour: If the sequence elements of become larger and larger (i.e. each subsequent sequence member $a_{n+1}$ is larger than $a_n$), 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:

Remark: convergent sequences

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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