A series is the sum of the terms of a sequence. For example, the harmonic series is
and the alternating harmonic series is
If we consider a general sequence
then the sum of the first terms is concisely denoted by .
When we encounter the expression we let the index ‘ ’ in the term take, in turn, the values and then add all these terms together. So, for example
Note that is a dummy index; any letter could be used as the index. For example , and each represent the same collection of terms: .
In order to be able to use this ‘summation notation’ we need to obtain a suitable expression for the ‘typical term’ in the series. For example, the finite series
may be written as since the typical term is clearly in which in turn.
In the same way
since an expression for the typical term in this alternating harmonic series is .
Write in summation form the series
First find an expression for the typical term, “the term”:
Now write the series in summation form:
Write out all the terms of the series .
Give the values in the typical term :