Series

Consider the following sequence of numbers (odd numbers):

1,3,5,...

What is the sum of the $3$ first terms of the sequence? Of course $1+3+5=9$ . But what is the sum of the $10\,000$ first terms of the sequence? This is more difficult to answer. Is there a formula that lets us determine this sum? This question leads to the so-called series.

Definition 1

Given a sequence $a$ :

(a_n)=(a_1,a_2, a_3, ..., a_n, ...)

Based on this sequence, a new sequence $s$ can be formed,

(s_n)=(s_1, s_2, s_3, ..., s_n, ...)

the so-called series of the sequence $a$ . The $n$ -th series element $s_n$ is defined as the sum of the $n$ first terms of the sequence of $a$ :

\begin{array}{lll} s_1 &=& a_1\\ s_2 &=& a_1+a_2\\ s_3 &=& a_1+a_2+a_3\\ ... & &\\ s_n &=& a_1+a_2+a_3+...+a_n\\ \end{array}

Example 1

Determine $s_3$ of the sequence $(a_n)=1,1.2, 1.4, 1.5, ...$ .

Solution

$s_3=1+1.2+1.4=3.6$

Extra: Sigma notation

Before we go on, we want to introduce the sigma notation. It helps to avoid long expressions like

s_n=a_1+a_2+a_3+...+a_n

a more compact notation is introduced:

Definition 2

Given a sequence $a$ :

(a_n)=(a_1,a_2, a_3, ...)

The sigma notation

\Sigma_{k=u}^{v} a_k

means "add the terms $a_k$ , where $k$ goes from $u$ to $v$ , i.e. $a_u+...+a_v$ . The symbol $\Sigma$ (Greek S) stands for "sum".

Some examples:

Example 2

\begin{array}{lll} \Sigma_{k=3}^5 a_k &=& a_3+a_4+a_5\\ \Sigma_{k=1}^1 a_k &=& a_1\\ \Sigma_{k=1}^2 a_k &=& a_1+a_2\\ \end{array}

If we have an explicit rule, it can be used directly in the sigma notation. For example, if the sequence $a$ is given by $a_k=k^2$ (k=1,2,...), then we have

\Sigma_{k=1}^3 k^2 = 1^2+2^2+3^2=14

Exercise 1

Write as a sum without the $\Sigma$ :

$\Sigma_{k=2}^4 (2k-1)$
$\Sigma_{i=1}^3 \frac{1}{i}$
$\Sigma_{w=6}^8 (3-w)$

Solution

$\Sigma_{k=2}^4 (2k-1)=3+5+7$
$\Sigma_{i=1}^3 \frac{1}{i}=1+\frac{1}{2}+\frac{1}{3}$
$\Sigma_{w=6}^8 (3-w) = -3+(-4)+(-5)=-3-4-5$

Back to the series. With the help of the sigma notation, we can thus write for the $n$ -th term $s_n$ :

s_n=\Sigma_{k=1}^n a_k=a_1+...+a_n

Example 3

s_4=\Sigma_{k=1}^4 a_k=a_1+a_2+a_3+a_4