Sigma Notation and Riemann Sums

Read this section to learn about area. Work through practice problems 1-9.

Sigma Notation and Riemann Sums

One strategy for calculating the area of a region is to cut the region into simple shapes, calculate the area of each simple shape, and then add these smaller areas together to get the area of the whole region. We will use that approach, but it is useful to have a notation for adding a lot of values together: the sigma $(\Sigma)$ notation.

Summation Sigma notation A way to read the sigma notation
$1^{2}+2^{2}+3^{2}+4^{2}+5^{2}$ $\sum_{\mathrm{k}=1}^{5} \mathrm{k}^{2}$ the sum of $k$ squared for $k$ equals 1 to $k$ equals 5
$\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$ $\sum_{\mathrm{k}=3}^{7} \frac{1}{\mathrm{k}}$ the sum of 1 divided by $k$ for $k$ equals 3 to $k$ equals 7
$2^{0}+2^{1}+2^{2}+2^{3}+2^{4}+2^{5}$ $\sum_{j=0}^{5} 2^{j}$ the sum of 2 to the jth power for $j$ equals 0 to $j$ equals 5
$a_{2}+a_{3}+a_{4}+a_{5}+a_{6}+a_{7}$ $\sum_{\mathrm{i}=2}^{7} \mathrm{a}_{\mathrm{i}}$ the sum of a sub $i$ from $i$ equals 2 to $i$ equals 7

The variable (typically $i$, $j$, or $k$) used in the summation is called the counter or index variable. The function to the right of the sigma is called the summand, and the numbers below and above the sigma are called the lower and upper limits of the summation. (Fig. 1)

Practice 1: Write the summation denoted by each of the following:

$(a) \sum_{k=1}^{5} k^{3}$,        $(b) \sum_{j=2}^{7}(-1)^{j} \frac{1}{j}$,            $(c) \sum_{\mathrm{m}=0}^{4}(2 \mathrm{~m}+1).$

In practice, the sigma notation is frequently used with the standard function notation:

$\sum_{\mathrm{k}=1}^{3} \mathrm{f}(\mathrm{k}+2)=\mathrm{f}(1+2)+\mathrm{f}(2+2)+\mathrm{f}(3+2)=\mathrm{f}(3)+\mathrm{f}(4)+\mathrm{f}(5)$ and

$\sum_{\mathrm{i}=1}^{4} \mathrm{f}\left(\mathrm{x}_{\mathrm{i}}\right)=\mathrm{f}\left(\mathrm{x}_{1}\right)+\mathrm{f}\left(\mathrm{x}_{2}\right)+\mathrm{f}\left(\mathrm{x}_{3}\right)+\mathrm{f}\left(\mathrm{x}_{4}\right)$

$x$ $f(x)$ $g(x)$ $h(x)$
1 2 4 3
2 3 1 3
3 1 -2 3
4 0 3 3
5 3 5 3

Table 1

Example 1: Use the values in Table 1 to evaluate $\sum_{k=2}^{5} 2 \cdot f(k)$ and $\sum_{j=3}^{5}(5+f(j-2))$.

Solution: $\sum_{k=2}^{5} 2 \cdot f(k)=2 \cdot f(2)+2 \cdot f(3)+2 \cdot f(4)+2 \cdot f(5)=2 \cdot(3)+2 \cdot(1)+2 \cdot(0)+2 \cdot(3)=14$

$\begin{gathered} \sum_{j=3}^{5}(5+f(j-2))=(5+f(3-2))+(5+f(4-2))+(5+f(5-2))=(5+f(1))+(5+f(2))+(5+f(3)) \\ =(5+2)+(5+3)+(5+1)=21 \end{gathered}$

Practice 2: Use the values of $f$, $g$ and $h$ in Table 1 to evaluate the following:

(a) $\sum_{k=2}^{5} g(k)$     (b) $\sum_{j=1}^{4} h(j)$     (c) $\sum_{k=3}^{5}(g(k)+f(k-1))$

Example 2: For $f(x)=x^{2}+1$, evalutate $\sum_{\mathrm{k}=0}^{3} \mathrm{f}(\mathrm{k})$.

Solution: $\sum_{k=0}^{3} f(k)=f(0)+f(1)+f(2)+f(3)$
$=\left(0^{2}+1\right)+\left(1^{2}+1\right)+\left(2^{2}+1\right)+\left(3^{2}+1\right)=1+2+5+10=18$

Practice 3: For $g(x)=1 / x$, evaluate $\sum_{k=2}^{4} g(k)$ and $\sum_{\mathrm{k}=1}^{3} \mathrm{~g}(\mathrm{k}+1)$.

The summand does not have to contain the index variable explicitly: a sum from $\mathrm{k}=2$ to $\mathrm{k}=4$ of the constant function $f(k)=5$ can be written as

$\sum_{k=2}^{4} f(k)$ or $\sum_{k=2}^{4} 5=5+5+5=3 \cdot 5=15$. Similarly, $\sum_{k=3}^{7} 2=2+2+2+2+2=52=10$.

Since the sigma notation is simply a notation for addition, it has all of the familiar properties of addition

Summation Properties

Sum of Constants: $\sum_{k=1}^{\mathrm{n}} \mathrm{C}=\mathrm{C}+\mathrm{C}+\mathrm{C}+\ldots+\mathrm{C}(\mathrm{n} \text { terms })=\mathrm{n} \cdot \mathrm{C}$

Addition: $\sum_{\mathrm{k}=1}^{\mathrm{n}}\left(\mathrm{a}_{\mathrm{k}}+\mathrm{b}_{\mathrm{k}}\right)=\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{k}}+\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{b}_{\mathrm{k}}$

Subtraction: $\sum_{\mathrm{k}=1}^{\mathrm{n}}\left(\mathrm{a}_{\mathrm{k}}-\mathrm{b}_{\mathrm{k}}\right)=\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{k}}-\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{b}_{\mathrm{k}}$

Constant Multiple: $\sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{C}^{*} \mathrm{a}_{\mathrm{k}}=\mathrm{C} \cdot \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{a}_{\mathrm{k}}$

Problems 16 and 17 illustrate that similar patterns for sums of products and quotients are not valid.