The notions of sequences and series

Sequences and series: The notions of sequence and series

The notions of sequences and series

In all sorts of cases, we encounter number sequences. Suppose $\euro \, 1000$ is deposited into a savings account, and each year $0.2\%$ of interest is added. The amounts $1000$ , $1002$ , $1004.004$ , $\ldots$ , accrued in the respective years 1, 2, 3, $\ldots$ on the account, are an example of a sequence. Here is a mathematical definition of a sequence.

Sequences

A sequence is an ordered list of numbers $a_1$ , $a_2, \ldots$ with an index (subscript).

The number $a_k$ , in which $k=1$ , $2$ , $3, \ldots$ , is also called a term or the $k$ -th term of the sequence.

In this case, the term $a_1$ is called the first or initial term, since it's the term with which the sequence starts.

A sequence is infinite if $a_k$ is defined for each number $k\ge 1$ (the index).

A finite sequence is often written as $a_1$ , $a_2$ , $\ldots$ , $a_n$ , in which $n$ is the length of the sequence.

Examples

Here are some examples of sequences.

There are sequences with a fixed difference between two terms, for example:

$a_1=2, a_2=5, a_3=8, a_4=11, a_5=14, \ldots$

There also are sequences where the terms are multiplied by a fixed number, for example:

$a_1=2, a_2=6, a_3=18, a_4=54, a_5=162, \ldots$

But entirely different patterns in sequences are also possible, for example:

$a_1=1, a_2=1, a_3=2, a_4=3, a_5=5, \ldots$

Sequences can, for example, be defined by recursive formulas. This means that the next term can be determined by looking at the previous one. An example of a recursive formula is $a_n=2+a_{n-1}$ . If $a_1$ is given, then we can calculate each term from the sequence. If $a_1=2$ , then the sequence is as follows: $a_1=2, \enspace a_2=4,\enspace a_3=6,\enspace a_4=8,\enspace a_5=10,\enspace \ldots$

Sequences can also be described by a function notation. For example: $a_n = \dfrac{1}{2^n}$ $(n\ge1)$ . In this case, all terms can be calculated directly, you don't need the previous term. This sequence then starts with $a_1=\dfrac{1}{2},\enspace a_2=\dfrac{1}{4},\enspace a_3=\dfrac{1}{8},\enspace\ldots$ Another example is $a_n=\begin{cases} n+1 & 1\leq n \lt 4\\ n^2 & n\geq 4\end{cases}$ This sequence starts with $a_1=2,\enspace a_2=3, \enspace a_3=4, \enspace a_4=16, \enspace a_5=25, \enspace \ldots$

In short, the sequence $a_1$ , $a_2$ , $\ldots$ can also be indicated by $a$ . In that way, the sequence can be understood as a function $a$ which assigns a number $a(k) = a_k$ to each natural number $k$ .

Sometimes it's easier to start with $0$ . In that case, we're talking about a sequence $a_0$ , $a_1$ , $\ldots$ Here $a_0$ is the first term.

We will now investigate a special form of a sequence, a series.

Series

A series is a special sequence $b_1$ , $b_2, \ldots$ in which the terms $b_n$ are constructed from terms $a_k$ from another sequence in the following manner $b_n=\sum_{k=1}^na_k=a_1+a_2+\cdots\,+a_n{,}\qquad n=1, 2, 3, \ldots$

A term $b_n$ of a series of sequence $a$ , hence, adds the first $n$ terms of a sequence $a$ .

For example, if for sequence $a$ we have $a_1=1$ , $a_2=2$ , $a_3=7$ , $a_4=-3$ , $a_5=-2$ , $a_6=3$ , then the corresponding series starts with the following terms $\begin{array}{rcl}b_1 &=& 1\\ b_2&=&1+2= 3\\ b_3&=&1+2+7=10\\ b_4&=&1+2+7-3=7\\ b_5&=&1+2+7-3-2=5\\ b_6&=&1+2+7-3-2+3=8\\ \end{array}$

The summation sign $\sum$ indicates a summation of terms. In the definition of a series, these are the terms of the sequence $a$ . Under the summation sign, you can find the index, $k$ , with its starting point, $1$ , and at the top of the sign its endpoint, $n$ . This means we want to sum all terms with index $1$ up to $n$ . For example, if $n=3$ , we get $b_3=\sum_{k=1}^3 a_k = a_1 + a_2+a_3$ Note that the index $k$ is a so-called dummy variable, meaning that $k$ has no meaning outside the summation sign. We could also use other letters for the index, for instance, $i$ or $j$ : $b_n=\sum_{k=1}^n a_k=\sum_{i=1}^n a_i=\sum_{j=1}^n a_j \quad\text{and}\quad b_3=\sum_{i=1}^3 a_i = a_1 + a_2+a_3$ Other examples of the use of summation signs are the following: $\begin{array}{rcl} \displaystyle \sum_{i=3}^5 x_i &=& x_3+x_4+x_5 \\ \displaystyle\sum_{k=1}^4 k &=& 1+2+3+4 \enspace=\enspace 10\\ \displaystyle \sum_{j=0}^3 (j^2+1) &=& (0^2+1)+(1^2+1)+(2^2+1)+(3^2+1) \enspace = \enspace 1+2+5+10 \enspace = \enspace 18 \\\displaystyle \sum_{k=1}^3 5 &=& 5+5+5 \enspace = \enspace 15\end{array}$

Since a series is a special sequence, all remarks in this chapter about sequences are also valid for series.

We will now take a look at some examples of sequences and series.

Consider the sequence $a_k=k$ , or $a_1=1{\tiny,}\quad a_2=2{\tiny,}\quad a_3=3{\tiny,}\quad\ldots$ What does the series $b_n=a_1+a_2+\cdots + a_n$ look like?

$b_1=1$ , $b_2=3$ , $b_3=6$ , $b_4=10$ , $b_5=15$ , $\ldots$

For, the first terms of the series $b$ can be calculated: $b_1=1$ , $b_2=1+2$ , $b_3=1+2+3$ , $b_4=1+2+3+4$ , $b_5=1+2+3+4+5$ , $\ldots$ This determines the correct answer.

By the way, the general term of the series is $b_n=\frac{1}{2}n\cdot(n+1)$ .

New example