3 Series

A series is the sum of the terms of a sequence. For example, the harmonic series is

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$

and the alternating harmonic series is

$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$

3.1 The summation notation

If we consider a general sequence

$a_1,a_2,\dots ,a_n,\dots$

then the sum of the first $k$ terms $a_1+a_2+a_3+\cdots +a_k$ is concisely denoted by ${\sum }_{p=1}^k{a}_p$ .

That is,

$a_1+a_2+a_3+\cdots +a_k={\sum }_{p=1}^k{a}_p$

When we encounter the expression ${\sum }_{p=1}^k{a}_p$ we let the index ‘ $p$ ’ in the term $a_p$ take, in turn, the values $1,2,\dots ,k$ and then add all these terms together. So, for example

${\sum }_{p=1}^3{a}_p=a_1+a_2+a_3{\sum }_{p=2}^7{a}_p=a_2+a_3+a_4+a_5+a_6+a_7$

Note that $p$ is a dummy index; any letter could be used as the index. For example ${\sum }_{i=1}^6{a}_i$ , and ${\sum }_{m=1}^6{a}_m$ each represent the same collection of terms: $a_1+a_2+a_3+a_4+a_5+a_6$ .

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

$1^2+2^2+\cdots +k^2$

may be written as ${\sum }_{p=1}^k{p}^2$ since the typical term is clearly $p^2$ in which $p=1,2,3,\dots ,k$ in turn.

In the same way

$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots -\frac{1}{16}={\sum }_{p=1}^{16}\frac{{\left(-1\right)}^{p+1}}{p}$

since an expression for the typical term in this alternating harmonic series is $a_p=\frac{{\left(-1\right)}^{p+1}}{p}$ .

Task!

Write in summation form the series

$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\cdots +\frac{1}{21\times 22}$

First find an expression for the typical term, “the $p^{th}$ term”:

Answer

Now write the series in summation form:

Answer

Write out all the terms of the series ${\sum }_{p=1}^5\frac{{\left(-1\right)}^p}{{\left(p+1\right)}^2}$ .

Give $p$ the values $1,2,3,4,5$ in the typical term $\frac{{\left(-1\right)}^p}{{\left(p+1\right)}^2}$ :

Answer