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.
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.
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.
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)#.
The notions of sequences and series
