3.4 Summation Notation

Let $k$ and $n$ be integers with $k \leq n$. Let $x_k, x_{k+1}, \ldots x_n$, be real numbers, indexed by the integers from $k$ to $n$. We define

$$\index{summation notation} \sum_{i=k}^n x_i = x_k + x_{k+1} + \cdots + x_n,$$ (3.53)

i.e. $${ \sum_{i=k}^n x_i }$$ is the sum of all the numbers $x_k,\ldots x_n$. A sum of one number is defined to be that number, so that

$$\sum_{i=k}^k x_i = x_k.$$

The ``$i$'' in equation (3.53) is a dummy variable, and can be replaced by any symbol that has no meaning assigned to it. Thus

$$\sum_{j=2}^4\frac{1}{j} = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{12}.$$

The following properties of the summation notation should be clear from the definition. (Here $c \in \mbox{{\bf R}}$, $k$ and $n$ are integers with $k \leq n$ and $x_k,\ldots,x_n$ and $y_k,\ldots y_n$ are real numbers.)

$$\begin{eqnarray*} \sum_{j=k}^n x_j + \sum_{j=k}^n y_j & = & \sum_{j=k}^n (x_j + ... ...eft( \sum_{j=k}^n x_j\right) + x_{n+1} &=& \sum_{j=k}^{n+1} x_j. \end{eqnarray*}$$

If $x_j \leq y_j$ for all $j$ satisfying $k \leq j \leq n$ then

$$\sum_{j=k}^n x_j \leq \sum_{j=k}^n y_j.$$

Also

$$\sum_{j=k}^n x_j = \sum_{j=k-1}^{n-1} x_{j+1} = \sum_{j=k+1}^{n+1} x_{j-1} = x_k + \cdots + x_n.$$

Using the summation notation, we can rewrite equations (2.9) and (2.23) as

$$\sum_{p=1}^n p^2 = \frac{n(n+1)(2n+1)}{6}$$

and

$$\sum_{j=0}^{n-1} r^j = \frac{1-r^n}{1-r}.$$

The use of the Greek letter $\Sigma$ to denote sums was introduced by Euler in 1755[15, page 61]. Euler writes

$$\Sigma x^2 = {x^3 \over 3} -{x^2 \over 2} + {x\over 6}.$$

Compare this with the notation in Bernoulli's table 2.2. (The apparent difference is due to the fact that for Euler, $\Sigma x^2$ denotes the sum of $x$ squares, starting with $0^2$, whereas for Bernoulli $\int nn$ denotes the sum of $n$ squares starting with $1^2$.) The use of the symbol $\int$ (which is a form of $S$) for sums was introduced by Leibniz. The use of limits on sums was introduced by Augustin Cauchy(1789-1857). Cauchy used the notation $${ \sum_m^n \!\!\!\! \mbox{\scriptsize$r$} \; fr}$$ to denote what we would write as $${\sum_{r=m}^n f(r)}$$[15, page 61].

3.54   Exercise. Find the following sums:

a)

$${ \sum_{j=1}^n (2j-1) \mbox{ for }n = 1,2,3,4.}$$

b)

$${ \sum_{j=1}^n {1\over j(j+1)} \mbox{ for }n = 1,2,3,4}.$$

c)

$${ \sum_{j=1}^9 {9 \over 10^j}.}$$