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In this section we simply work on the concept of adding up the numbers belonging to arithmetic and geometric sequences. We call the sum of any sequence of numbers a series .
If we add up the terms of a sequence, we obtain what is called a series . If we only sum a finite amount of terms, we get a finite series . We use the symbol ${S}_{n}$ to mean the sum of the first $n$ terms of a sequence $\{{a}_{1};{a}_{2};{a}_{3};...;{a}_{n}\}$ :
For example, if we have the following sequence of numbers
and we wish to find the sum of the first 4 terms, then we write
The above is an example of a finite series since we are only summing 4 terms.
If we sum infinitely many terms of a sequence, we get an infinite series :
In this section we introduce a notation that will make our lives a little easier.
A sum may be written out using the summation symbol $\sum $ . This symbol is sigma , which is the capital letter “S” in the Greek alphabet. It indicates that you must sum the expression to the right of it:
where
The index $i$ is increased from $m$ to $n$ in steps of 1.
If we are summing from $n=1$ (which implies summing from the first term in a sequence), then we can use either ${S}_{n}$ - or $\sum $ -notation since they mean the same thing:
For example, in the following sum,
we have to add together all the terms in the sequence ${a}_{i}=i$ from $i=1$ up until $i=5$ :
Remember that an arithmetic sequence is a set of numbers, such that the difference between any term and the previous term is a constant number, $d$ , called the constant difference :
where
When we sum a finite number of terms in an arithmetic sequence, we get a finite arithmetic series .
The simplest arithmetic sequence is when ${a}_{1}=1$ and $d=0$ in the general form [link] ; in other words all the terms in the sequence are 1:
If we wish to sum this sequence from $i=1$ to any positive integer $n$ , we would write
Since all the terms are equal to 1, it means that if we sum to $n$ we will be adding $n$ -number of 1's together, which is simply equal to $n$ :
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